Fixed-Factor Exact Discrete Exponential Models
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Discrete exponential growth model · qf-app20-growth-factor-regime · candidate
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