# Question 5408: Inclusive Finite Recurrence Prefix [Return to catalogue page 28](https://ocme.evemisslab.com/en/questions/page/28/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261012-r01-mathematical-induction-foundations/#q-20261008-advance-20261012-r01-mathematical-induction-0033) 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. R0009-05408 · q-20261008-advance-20261012-r01-mathematical-induction-0033 ## Original question and givens N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-5, a(k+1)=a(k)+(0), through=8. List a(0) through a(through), inclusive. - G001 — N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-5, a(k+1)=a(k)+(0), through=8. List a(0) through a(through), inclusive. ## 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-mathematical-induction](https://ocme.evemisslab.com/en/mko/mko-mathematical-induction/) ## Step-by-step derivation S001 · impliesBefore natural indices start at0;finite induction protocol After index bookkeeping and finite values are not proofs of all P(n) Conditions: none Natural indices do not require nonnegative sequence values.Checking a finite range or a successor index does not supply proposition truth or an induction proof over all naturals. Step identifiers and dependenciesRule: finite_induction_claim_boundary · Uses: G001 S002 · impliesBefore a(0)=-5;through=8 After [-5] Conditions: none Start with the supplied initial value;through=0 retains only it. Step identifiers and dependenciesRule: recurrence_prefix_base · Uses: G001, S001 S003 · calculationBefore recurrence next value1 := sum(-5, 0) After recurrence next value1 = -5 Conditions: none Add the declared d to the previous value;negative increments or terms are allowed. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore a(0)=-5;d=0 After a(1)=-5 Conditions: none Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix. Step identifiers and dependenciesRule: finite_recurrence_transition · Uses: G001, S003 S005 · calculationBefore recurrence next value2 := sum(-5, 0) After recurrence next value2 = -5 Conditions: none Add the declared d to the previous value;negative increments or terms are allowed. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore a(1)=-5;d=0 After a(2)=-5 Conditions: none Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix. Step identifiers and dependenciesRule: finite_recurrence_transition · Uses: G001, S005 S007 · calculationBefore recurrence next value3 := sum(-5, 0) After recurrence next value3 = -5 Conditions: none Add the declared d to the previous value;negative increments or terms are allowed. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore a(2)=-5;d=0 After a(3)=-5 Conditions: none Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix. Step identifiers and dependenciesRule: finite_recurrence_transition · Uses: G001, S007 S009 · calculationBefore recurrence next value4 := sum(-5, 0) After recurrence next value4 = -5 Conditions: none Add the declared d to the previous value;negative increments or terms are allowed. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore a(3)=-5;d=0 After a(4)=-5 Conditions: none Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix. Step identifiers and dependenciesRule: finite_recurrence_transition · Uses: G001, S009 S011 · calculationBefore recurrence next value5 := sum(-5, 0) After recurrence next value5 = -5 Conditions: none Add the declared d to the previous value;negative increments or terms are allowed. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore a(4)=-5;d=0 After a(5)=-5 Conditions: none Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix. Step identifiers and dependenciesRule: finite_recurrence_transition · Uses: G001, S011 S013 · calculationBefore recurrence next value6 := sum(-5, 0) After recurrence next value6 = -5 Conditions: none Add the declared d to the previous value;negative increments or terms are allowed. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore a(5)=-5;d=0 After a(6)=-5 Conditions: none Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix. Step identifiers and dependenciesRule: finite_recurrence_transition · Uses: G001, S013 S015 · calculationBefore recurrence next value7 := sum(-5, 0) After recurrence next value7 = -5 Conditions: none Add the declared d to the previous value;negative increments or terms are allowed. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore a(6)=-5;d=0 After a(7)=-5 Conditions: none Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix. Step identifiers and dependenciesRule: finite_recurrence_transition · Uses: G001, S015 S017 · calculationBefore recurrence next value8 := sum(-5, 0) After recurrence next value8 = -5 Conditions: none Add the declared d to the previous value;negative increments or terms are allowed. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore a(7)=-5;d=0 After a(8)=-5 Conditions: none Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix. Step identifiers and dependenciesRule: finite_recurrence_transition · Uses: G001, S017 ## Reverse checks - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute recurrence next value8 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute recurrence next value7 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Recompute recurrence next value6 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute recurrence next value5 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute recurrence next value4 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute recurrence next value3 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 — Recompute recurrence next value2 with exact rational arithmetic, not rounded decimals. - 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 — Recompute recurrence next value1 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 [-5, -5, -5, -5, -5, -5, -5, -5, -5] 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: f73ac7eec4b420ad7028b22e9375d16d22384627625e8a1aa7e06fef167ecc10 - candidate record SHA-256: da9e8cbfbdea8d758da475d2172715e4f0be5109be209a4c81ff8c81872a606f - semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56 - question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r01-mathematical-induction-0033.json)