# Question 35091: Term of a Zero-Based Explicit Sequence [Return to catalogue page 176](https://ocme.evemisslab.com/en/questions/page/176/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260930-r05-sequence-foundations/#q-20260930-r05-sequence-explicit-term-0016) 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. R0069-35091 · q-20260930-r05-sequence-explicit-term-0016 ## Original question and givens The real sequence is indexed from 0 and has the explicit rule a_n=(3)+(-2)n for n∈N={0,1,2,…}. Find the term at n=5. - G001 — The real sequence is indexed from 0 and has the explicit rule a_n=(3)+(-2)n for n∈N={0,1,2,…}. Find the term at n=5. ## 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-explicit-term, [d, n, operation, u]] After derived = [explicit_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 = [3, -2, 5, explicit_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 S003 · calculationBefore Sequence ORIGINAL zero-based n times common difference := mul(5, -2) After Sequence ORIGINAL zero-based n times common difference = -10 Conditions: none Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Sequence ORIGINAL initial plus n common differences := sum(3, -10) After Sequence ORIGINAL initial plus n common differences = -7 Conditions: none Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [3, -2, 5] After derived = [-7] Conditions: none Substitute THIS n into u+n*d.At n=0 recover a0=u;negative d is allowed.Derived BigInts are not recapped at RAW SAFE;one term does not prove a limit. Step identifiers and dependenciesRule: Sequence_EXPLICIT_complete_substitution_NO_limit_claim · Uses: G001, S004 ## Reverse checks - S005 · 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. - S004 · premise_and_result_audit — Recompute Sequence ORIGINAL initial plus n common differences with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Sequence ORIGINAL zero-based n times common difference with exact rational arithmetic, not rounded decimals. - 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 -7 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: 068cc86fc833ae84637eeaf77f1ef67d496d15d5992daf8095dcd62e7765a127 - candidate record SHA-256: d0f5b9fa9da25a63b6820d1b457ac1a953fbe79a1c923bb8510d8ce9fe2a594e - 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-explicit-term-0016.json)