# Question 1825: Exact Fixed-Period Compound Value Including Zero Periods [Return to catalogue page 10](https://ocme.evemisslab.com/en/questions/page/10/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261020-r07-compound-interest-foundations/#q-20261013-advance-20261020-r07-compound-interest-0050) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0002-01825 · q-20261013-advance-20261020-r07-compound-interest-0050 ## Original question and givens Use one fixed exact rate per declared period and no deposits, withdrawals, fees, taxes or rounding. Rate and period conventions agree; balances/principal are nonnegative and 1+rate>0. This is not investment advice, a quote or a market forecast. 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. principal=220/1, rate=2/20, periods=0. Give (principal, rate, accumulation factor, periods, exact future value). - G001 — Use one fixed exact rate per declared period and no deposits, withdrawals, fees, taxes or rounding. Rate and period conventions agree; balances/principal are nonnegative and 1+rate>0. This is not investment advice, a quote or a market forecast. 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. principal=220/1, rate=2/20, periods=0. Give (principal, rate, accumulation factor, periods, exact future value). ## 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-compound-interest](https://ocme.evemisslab.com/en/mko/mko-compound-interest/) ## Step-by-step derivation S001 · calculationBefore accumulation factor := sum(1, 1/10) After accumulation factor = 11/10 Conditions: none Add1 to the per-period rate to obtain the multiplicative accumulation factor. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore per-period rate=1/10; factor=11/10>0 After fixed-period exact recurrence B_(k+1)=(1+r)B_k Conditions: none Use only the exact fixed-period,fixed-rate model with no extra cash flows;this is not product advice and no currency rounding is introduced. Step identifiers and dependenciesRule: compound_model_assumptions · Uses: G001, S001 S003 · impliesBefore B₀=220; periods=0 After 0 periods leave the principal unchanged Conditions: none At0 periods the balance is the principal;do not add a period of interest. Step identifiers and dependenciesRule: principal_and_period_count · Uses: G001, S002 S004 · impliesBefore B₀=P; B_(k+1)=(1+r)B_k After B_n=P(1+r)^n by induction Conditions: none The0-period base case holds,and multiplying by the same factor proves the next induction case. Step identifiers and dependenciesRule: compound_power_identity · Uses: G001, S003 S005 · impliesBefore (11/10)^0 After (1+r)^n = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S004 S006 · calculationBefore future balance := mul(220, 1) After future balance = 220 Conditions: none Substitute into the power formula and cross-check against the periodwise balances. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Recompute future balance with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute accumulation factor with exact rational arithmetic, not rounded decimals. ## Result (220, 1/10, 11/10, 0, 220) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 3671dd5f80803980d3871df5afc33dee98e501a15c309c360d9b14f5caeb0f01 - candidate record SHA-256: bdafa1cf1eb125513204cab21a18557760e17876f4483077f9832a85802b05b1 - semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861 - question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261020-r07-compound-interest-0050.json)