Question 1758: The Exact Natural-Indexed Term Including Zero Factor and Time

Original question and givens

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.

• G001 — 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.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-discrete-exponential-growth

Step-by-step derivation

factor q=4/3≥0

X_(k+1)=q·X_k

X₀=-1/3; n=2

n=0 uses the initial value; an empty product equals1

X_1 := mul(-1/3, 4/3)

X_1 = -4/9

X_2 := mul(-4/9, 4/3)

X_2 = -16/27

X_k=X₀q^k ⇒ X_(k+1)=qX_k=X₀q^(k+1)

X_n=X₀q^n for every nonnegative integer n

(4/3)^2

q^n = 16/9

closed-form X_n := mul(-1/3, 16/9)

closed-form X_n = -16/27

Reverse checks

• S007 · premise_and_result_audit — Recompute closed-form X_n with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute X_2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute X_1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(-1/3, 4/3, 2, -16/27)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.