# Fixed-Factor Exact Discrete Exponential Models

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-8/1, factor=1/2. Give (current, factor, next=current*factor).

View answer and explanation(-8, 1/2, -4)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-8, 1/2, -4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0001/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 2. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-7/2, factor=2/3. Give (current, factor, next=current*factor).

View answer and explanation(-7/2, 2/3, -7/3)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-7/2, 2/3, -7/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0002/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 3. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-6/3, factor=3/4. Give (current, factor, next=current*factor).

View answer and explanation(-2, 3/4, -3/2)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-2, 3/4, -3/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0003/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 4. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-5/4, factor=4/5. Give (current, factor, next=current*factor).

View answer and explanation(-5/4, 4/5, -1)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-5/4, 4/5, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0004/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 5. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-4/5, factor=5/6. Give (current, factor, next=current*factor).

View answer and explanation(-4/5, 5/6, -2/3)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-4/5, 5/6, -2/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0005/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 6. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-3/1, factor=6/7. Give (current, factor, next=current*factor).

View answer and explanation(-3, 6/7, -18/7)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-3, 6/7, -18/7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0006/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 7. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-2/2, factor=7/2. Give (current, factor, next=current*factor).

View answer and explanation(-1, 7/2, -7/2)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-1, 7/2, -7/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0007/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 8. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-1/3, factor=1/3. Give (current, factor, next=current*factor).

View answer and explanation(-1/3, 1/3, -1/9)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-1/3, 1/3, -1/9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0008/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 9. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=0/4, factor=2/4. Give (current, factor, next=current*factor).

View answer and explanation(0, 1/2, 0)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (0, 1/2, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0009/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 10. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=1/5, factor=3/5. Give (current, factor, next=current*factor).

View answer and explanation(1/5, 3/5, 3/25)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/5, 3/5, 3/25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0010/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 11. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=2/1, factor=4/6. Give (current, factor, next=current*factor).

View answer and explanation(2, 2/3, 4/3)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, 2/3, 4/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0011/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 12. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=3/2, factor=5/7. Give (current, factor, next=current*factor).

View answer and explanation(3/2, 5/7, 15/14)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3/2, 5/7, 15/14).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0012/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 13. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=4/3, factor=6/2. Give (current, factor, next=current*factor).

View answer and explanation(4/3, 3, 4)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4/3, 3, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0013/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 14. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=5/4, factor=7/3. Give (current, factor, next=current*factor).

View answer and explanation(5/4, 7/3, 35/12)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/4, 7/3, 35/12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0014/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 15. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=6/5, factor=1/4. Give (current, factor, next=current*factor).

View answer and explanation(6/5, 1/4, 3/10)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6/5, 1/4, 3/10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0015/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 16. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=7/1, factor=2/5. Give (current, factor, next=current*factor).

View answer and explanation(7, 2/5, 14/5)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 2/5, 14/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0016/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 17. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=8/2, factor=3/6. Give (current, factor, next=current*factor).

View answer and explanation(4, 1/2, 2)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4, 1/2, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0017/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 18. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=9/3, factor=4/7. Give (current, factor, next=current*factor).

View answer and explanation(3, 4/7, 12/7)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, 4/7, 12/7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0018/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 19. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=10/4, factor=5/2. Give (current, factor, next=current*factor).

View answer and explanation(5/2, 5/2, 25/4)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/2, 5/2, 25/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0019/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 20. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=11/5, factor=6/3. Give (current, factor, next=current*factor).

View answer and explanation(11/5, 2, 22/5)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (11/5, 2, 22/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0020/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 21. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=12/1, factor=7/4. Give (current, factor, next=current*factor).

View answer and explanation(12, 7/4, 21)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (12, 7/4, 21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0021/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 22. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=13/2, factor=1/5. Give (current, factor, next=current*factor).

View answer and explanation(13/2, 1/5, 13/10)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (13/2, 1/5, 13/10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0022/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 23. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=14/3, factor=2/6. Give (current, factor, next=current*factor).

View answer and explanation(14/3, 1/3, 14/9)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (14/3, 1/3, 14/9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0023/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 24. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=15/4, factor=3/7. Give (current, factor, next=current*factor).

View answer and explanation(15/4, 3/7, 45/28)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (15/4, 3/7, 45/28).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0024/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 25. One Exact Fixed-Factor Discrete Update

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=16/5, factor=4/2. Give (current, factor, next=current*factor).

View answer and explanation(16/5, 2, 32/5)

Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (16/5, 2, 32/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0025/)

Discrete exponential growth model · qf-app20-growth-recurrence-step · candidate





## 26. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-8/1, factor=1/2, n=0≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(-8, 1/2, 0, -8)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-8, 1/2, 0, -8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0026/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 27. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-7/2, factor=2/3, n=1≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(-7/2, 2/3, 1, -7/3)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-7/2, 2/3, 1, -7/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0027/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 28. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-6/3, factor=3/4, n=2≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(-2, 3/4, 2, -9/8)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-2, 3/4, 2, -9/8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0028/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 29. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-5/4, factor=4/5, n=3≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(-5/4, 4/5, 3, -16/25)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-5/4, 4/5, 3, -16/25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0029/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 30. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-4/5, factor=1/6, n=4≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(-4/5, 1/6, 4, -1/1620)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-4/5, 1/6, 4, -1/1620).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0030/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 31. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-3/1, factor=2/7, n=0≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(-3, 2/7, 0, -3)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-3, 2/7, 0, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0031/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 32. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-2/2, factor=3/2, n=1≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(-1, 3/2, 1, -3/2)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-1, 3/2, 1, -3/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0032/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 33. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-1/3, factor=4/3, n=2≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(-1/3, 4/3, 2, -16/27)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-1/3, 4/3, 2, -16/27).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0033/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 34. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=0/4, factor=1/4, n=3≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(0, 1/4, 3, 0)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (0, 1/4, 3, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0034/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 35. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=1/5, factor=2/5, n=4≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(1/5, 2/5, 4, 16/3125)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/5, 2/5, 4, 16/3125).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0035/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 36. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=2/1, factor=3/6, n=0≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(2, 1/2, 0, 2)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, 1/2, 0, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0036/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 37. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=3/2, factor=4/7, n=1≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(3/2, 4/7, 1, 6/7)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3/2, 4/7, 1, 6/7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0037/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 38. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=4/3, factor=1/2, n=2≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(4/3, 1/2, 2, 1/3)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4/3, 1/2, 2, 1/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0038/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 39. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=5/4, factor=2/3, n=3≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(5/4, 2/3, 3, 10/27)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/4, 2/3, 3, 10/27).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0039/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 40. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=6/5, factor=3/4, n=4≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(6/5, 3/4, 4, 243/640)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6/5, 3/4, 4, 243/640).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0040/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 41. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=7/1, factor=4/5, n=0≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(7, 4/5, 0, 7)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 4/5, 0, 7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0041/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 42. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=8/2, factor=1/6, n=1≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(4, 1/6, 1, 2/3)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4, 1/6, 1, 2/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0042/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 43. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=9/3, factor=2/7, n=2≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(3, 2/7, 2, 12/49)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, 2/7, 2, 12/49).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0043/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 44. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=10/4, factor=3/2, n=3≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(5/2, 3/2, 3, 135/16)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/2, 3/2, 3, 135/16).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0044/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 45. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=11/5, factor=4/3, n=4≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(11/5, 4/3, 4, 2816/405)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (11/5, 4/3, 4, 2816/405).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0045/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 46. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=12/1, factor=1/4, n=0≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(12, 1/4, 0, 12)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (12, 1/4, 0, 12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0046/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 47. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=13/2, factor=2/5, n=1≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(13/2, 2/5, 1, 13/5)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (13/2, 2/5, 1, 13/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0047/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 48. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=14/3, factor=3/6, n=2≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(14/3, 1/2, 2, 7/6)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (14/3, 1/2, 2, 7/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0048/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 49. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=15/4, factor=4/7, n=3≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(15/4, 4/7, 3, 240/343)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (15/4, 4/7, 3, 240/343).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0049/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 50. The Exact Natural-Indexed Term Including Zero Factor and Time

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=16/5, factor=1/2, n=4≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.

View answer and explanation(16/5, 1/2, 4, 1/5)

Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (16/5, 1/2, 4, 1/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0050/)

Discrete exponential growth model · qf-app20-growth-nth-term · candidate





## 51. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-8/1, factor=1/2, n=0, candidate=-7/1. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(-8, 1/2, 0, -8, -7, No)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-8, 1/2, 0, -8, -7, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0051/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 52. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-7/2, factor=2/3, n=1, candidate=-7/3. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(-7/2, 2/3, 1, -7/3, -7/3, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-7/2, 2/3, 1, -7/3, -7/3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0052/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 53. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-2/1, factor=3/4, n=2, candidate=-9/8. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(-2, 3/4, 2, -9/8, -9/8, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-2, 3/4, 2, -9/8, -9/8, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0053/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 54. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-5/4, factor=4/5, n=3, candidate=-15/25. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(-5/4, 4/5, 3, -16/25, -3/5, No)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-5/4, 4/5, 3, -16/25, -3/5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0054/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 55. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-4/5, factor=1/6, n=4, candidate=-1/1620. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(-4/5, 1/6, 4, -1/1620, -1/1620, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-4/5, 1/6, 4, -1/1620, -1/1620, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0055/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 56. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-3/1, factor=2/7, n=0, candidate=-3/1. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(-3, 2/7, 0, -3, -3, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-3, 2/7, 0, -3, -3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0056/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 57. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-1/1, factor=3/2, n=1, candidate=-2/2. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(-1, 3/2, 1, -3/2, -1, No)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-1, 3/2, 1, -3/2, -1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0057/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 58. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-1/3, factor=4/3, n=2, candidate=-16/27. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(-1/3, 4/3, 2, -16/27, -16/27, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-1/3, 4/3, 2, -16/27, -16/27, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0058/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 59. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=0/1, factor=1/4, n=3, candidate=0/1. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(0, 1/4, 3, 0, 0, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (0, 1/4, 3, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0059/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 60. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=1/5, factor=2/5, n=4, candidate=17/3125. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(1/5, 2/5, 4, 16/3125, 17/3125, No)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/5, 2/5, 4, 16/3125, 17/3125, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0060/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 61. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=2/1, factor=1/2, n=0, candidate=2/1. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(2, 1/2, 0, 2, 2, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, 1/2, 0, 2, 2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0061/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 62. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=3/2, factor=4/7, n=1, candidate=6/7. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(3/2, 4/7, 1, 6/7, 6/7, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3/2, 4/7, 1, 6/7, 6/7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0062/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 63. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=4/3, factor=1/2, n=2, candidate=2/3. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(4/3, 1/2, 2, 1/3, 2/3, No)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4/3, 1/2, 2, 1/3, 2/3, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0063/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 64. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=5/4, factor=2/3, n=3, candidate=10/27. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(5/4, 2/3, 3, 10/27, 10/27, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/4, 2/3, 3, 10/27, 10/27, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0064/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 65. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=6/5, factor=3/4, n=4, candidate=243/640. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(6/5, 3/4, 4, 243/640, 243/640, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6/5, 3/4, 4, 243/640, 243/640, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0065/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 66. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=7/1, factor=4/5, n=0, candidate=8/1. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(7, 4/5, 0, 7, 8, No)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 4/5, 0, 7, 8, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0066/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 67. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=4/1, factor=1/6, n=1, candidate=2/3. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(4, 1/6, 1, 2/3, 2/3, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4, 1/6, 1, 2/3, 2/3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0067/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 68. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=3/1, factor=2/7, n=2, candidate=12/49. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(3, 2/7, 2, 12/49, 12/49, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, 2/7, 2, 12/49, 12/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0068/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 69. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=5/2, factor=3/2, n=3, candidate=136/16. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(5/2, 3/2, 3, 135/16, 17/2, No)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/2, 3/2, 3, 135/16, 17/2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0069/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 70. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=11/5, factor=4/3, n=4, candidate=2816/405. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(11/5, 4/3, 4, 2816/405, 2816/405, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (11/5, 4/3, 4, 2816/405, 2816/405, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0070/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 71. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=12/1, factor=1/4, n=0, candidate=12/1. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(12, 1/4, 0, 12, 12, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (12, 1/4, 0, 12, 12, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0071/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 72. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=13/2, factor=2/5, n=1, candidate=14/5. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(13/2, 2/5, 1, 13/5, 14/5, No)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (13/2, 2/5, 1, 13/5, 14/5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0072/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 73. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=14/3, factor=1/2, n=2, candidate=7/6. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(14/3, 1/2, 2, 7/6, 7/6, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (14/3, 1/2, 2, 7/6, 7/6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0073/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 74. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=15/4, factor=4/7, n=3, candidate=240/343. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(15/4, 4/7, 3, 240/343, 240/343, Yes)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (15/4, 4/7, 3, 240/343, 240/343, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0074/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 75. Normalize a Candidate before a Fixed-Factor Closed-Form Audit

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=16/5, factor=1/2, n=4, candidate=2/5. Give (initial, factor, n, exact term, normalized candidate, equal?).

View answer and explanation(16/5, 1/2, 4, 1/5, 2/5, No)

Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (16/5, 1/2, 4, 1/5, 2/5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0075/)

Discrete exponential growth model · qf-app20-growth-closed-form-audit · candidate





## 76. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=0/1. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(0, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (0, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0076/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 77. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=1/2. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(1/2, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/2, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0077/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 78. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=2/3. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(2/3, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2/3, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0078/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 79. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=3/4. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(3/4, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3/4, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0079/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 80. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=4/5. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(4/5, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4/5, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0080/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 81. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=5/6. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(5/6, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/6, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0081/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 82. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=6/7. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(6/7, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6/7, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0082/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 83. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=7/1. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(7, factor greater than 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, factor greater than 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0083/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 84. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=8/2. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(4, factor greater than 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4, factor greater than 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0084/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 85. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=0/3. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(0, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (0, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0085/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 86. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=1/4. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(1/4, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/4, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0086/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 87. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=2/5. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(2/5, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2/5, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0087/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 88. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=3/6. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(1/2, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/2, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0088/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 89. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=4/7. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(4/7, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4/7, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0089/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 90. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=5/1. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(5, factor greater than 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, factor greater than 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0090/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 91. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=6/2. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(3, factor greater than 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, factor greater than 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0091/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 92. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=7/3. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(7/3, factor greater than 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7/3, factor greater than 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0092/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 93. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=8/4. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(2, factor greater than 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, factor greater than 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0093/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 94. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=0/5. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(0, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (0, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0094/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 95. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=1/6. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(1/6, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/6, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0095/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 96. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=2/7. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(2/7, factor at least 0 and below 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2/7, factor at least 0 and below 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0096/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 97. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=3/1. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(3, factor greater than 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, factor greater than 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0097/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 98. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=4/2. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(2, factor greater than 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, factor greater than 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0098/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 99. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=5/3. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(5/3, factor greater than 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/3, factor greater than 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0099/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate





## 100. Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity

Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=6/4. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.

View answer and explanation(3/2, factor greater than 1)

Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3/2, factor greater than 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0100/)

Discrete exponential growth model · qf-app20-growth-factor-regime · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations/validation-report.json)