Explore related knowledge and questions →
← Question derivation index · Original question batch
R0002-01814 · q-20261013-advance-20261020-r07-compound-interest-0039
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=165/2, rate=1/20, periods=1. 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=165/2, rate=1/20, periods=1. Give (principal, rate, accumulation factor, periods, exact future value).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.
Before
accumulation factor := sum(1, 1/20)
After
accumulation factor = 21/20
Conditions: none
Add1 to the per-period rate to obtain the multiplicative accumulation factor.
Rule: exact_rational_arithmetic · Uses: G001
Before
per-period rate=1/20; factor=21/20>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.
Rule: compound_model_assumptions · Uses: G001, S001
Before
B₀=165/2; periods=1
After
0 periods leave the principal unchanged
Conditions: none
At0 periods the balance is the principal;do not add a period of interest.
Rule: principal_and_period_count · Uses: G001, S002
Before
B_1 := mul(165/2, 21/20)
After
B_1 = 693/8
Conditions: none
Accumulate into the balance before the next period,using the same factor each time.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
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.
Rule: compound_power_identity · Uses: G001, S004
Before
(21/20)^1
After
(1+r)^n = 21/20
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S005
Before
future balance := mul(165/2, 21/20)
After
future balance = 693/8
Conditions: none
Substitute into the power formula and cross-check against the periodwise balances.
Rule: exact_rational_arithmetic · Uses: G001, S006
S007 · premise_and_result_audit — Recompute future balance 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 B_1 with exact rational arithmetic, not rounded decimals.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.(165/2, 1/20, 21/20, 1, 693/8)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: ac1ddfb426118c6d11251246ba4836a4754429ba34a7da7f9203b42e06bd0582candidate record SHA-256: 19e554e6dedb75db00dd79da35301ab6df1efdd3da650a918b21bd521a6302f9semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED