# Question 35151: Term of a Recurrence with an Initial Value

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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.

R0069-35151 · q-20260930-r05-sequence-recursive-term-0001



## Original question and givens

The real sequence is indexed from 0 with a_0=12 and recurrence a_(j+1)=a_j+(-3) for every j≥0. Find the term at n=0.


- G001 — The real sequence is indexed from 0 with a_0=12 and recurrence a_(j+1)=a_j+(-3) for every j≥0. Find the term at n=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-sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-sequence-recursive-term, [d, n, operation, u]]

After

derived = [recursive_term, no-invented-domain]

Conditions: none

Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract.

Step identifiers and dependenciesRule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001





S002 · impliesBefore

source roles = [12, -3, 0, recursive_term]

After

derived = [a_0=u, a_n=u+n*d, zero based]

Conditions: none

Type ALL RAW u/d/count fields EVEN for prefix L=0.Indices start at0 and the formula/recurrence is GIVEN;do not infer a general rule from finite samples.

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









## Reverse checks


- S002 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S001 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.





## Result

12

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: 3d81cd93c9a7de13d29129160c37ceebaf34c06dec8ea42780f64145b1add925
- candidate record SHA-256: 444ed4dccdab034192f94fd6a76c87a79283de809249fb8a46d8fa7dae00b637
- semantic review receipt SHA-256: ccf93c72638d6abf6a412514da5838d55ecb09b71d0e4fdec2b559dc9fdee14b
- question-first witness SHA-256: 11af9a338b5a610342b2cfb2ee295683131b3139ac402dd19b1ba1725976d518
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260930-r05-sequence-recursive-term-0001.json)