# Question 5401: Inclusive Finite Recurrence Prefix

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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.

R0009-05401 · q-20261008-advance-20261012-r01-mathematical-induction-0026



## Original question and givens

N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-2, a(k+1)=a(k)+(99), through=0. 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)=-2, a(k+1)=a(k)+(99), through=0. 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)=-2;through=0

After

[-2]

Conditions: none

Start with the supplied initial value;through=0 retains only it.

Step identifiers and dependenciesRule: recurrence_prefix_base · Uses: G001, S001









## Reverse checks


- 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

[-2]

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: 2f4849764c6e263239583423d8d31d95db30402f3dda167e4e3f5be3d11a6258
- candidate record SHA-256: 9298aa01c30e3d611a7b44653c1fce4aae391c76a8a01a9a6c80f3b7efec6600
- 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-0026.json)