# Question 35147: Finite Prefix of an 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-prefix-0022) 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. R0069-35147 · q-20260930-r05-sequence-prefix-0022 ## Original question and givens The real sequence is indexed from 0 with a_n=(9)+(-3)n. List the prefix [a_0,…,a_(L-1)] of length L=6; use [] when L=0. - G001 — The real sequence is indexed from 0 with a_n=(9)+(-3)n. List the prefix [a_0,…,a_(L-1)] of length L=6; use [] when L=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-prefix, [L, d, operation, u]] After derived = [prefix, 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 = [9, -3, 6, prefix] 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 prefix ORIGINAL index times step j=0 := mul(0, -3) After Sequence prefix ORIGINAL index times step j=0 = 0 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 prefix ORIGINAL complete term j=0 := sum(9, 0) After Sequence prefix ORIGINAL complete term j=0 = 9 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 · calculationBefore Sequence prefix ORIGINAL index times step j=1 := mul(1, -3) After Sequence prefix ORIGINAL index times step j=1 = -3 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, S004 S006 · calculationBefore Sequence prefix ORIGINAL complete term j=1 := sum(9, -3) After Sequence prefix ORIGINAL complete term j=1 = 6 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, S005 S007 · calculationBefore Sequence prefix ORIGINAL index times step j=2 := mul(2, -3) After Sequence prefix ORIGINAL index times step j=2 = -6 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, S006 S008 · calculationBefore Sequence prefix ORIGINAL complete term j=2 := sum(9, -6) After Sequence prefix ORIGINAL complete term j=2 = 3 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, S007 S009 · calculationBefore Sequence prefix ORIGINAL index times step j=3 := mul(3, -3) After Sequence prefix ORIGINAL index times step j=3 = -9 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, S008 S010 · calculationBefore Sequence prefix ORIGINAL complete term j=3 := sum(9, -9) After Sequence prefix ORIGINAL complete term j=3 = 0 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, S009 S011 · calculationBefore Sequence prefix ORIGINAL index times step j=4 := mul(4, -3) After Sequence prefix ORIGINAL index times step j=4 = -12 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, S010 S012 · calculationBefore Sequence prefix ORIGINAL complete term j=4 := sum(9, -12) After Sequence prefix ORIGINAL complete term j=4 = -3 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, S011 S013 · calculationBefore Sequence prefix ORIGINAL index times step j=5 := mul(5, -3) After Sequence prefix ORIGINAL index times step j=5 = -15 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, S012 S014 · calculationBefore Sequence prefix ORIGINAL complete term j=5 := sum(9, -15) After Sequence prefix ORIGINAL complete term j=5 = -6 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, S013 S015 · impliesBefore source roles = [9, -3, 6] After derived = [[9, 6, 3, 0, -3, -6], [9, 6, 3, 0, -3, -6]] Conditions: none Take EXACTLY L terms in index order from a0 through a_(L−1),NOT through aL.L=0 displays[].The GIVEN formula determines these terms;the prefix alone need not determine an unknown rule. Step identifiers and dependenciesRule: Sequence_PREFIX_EXACT_length_L_indices0_to_Lminus1 · Uses: G001, S014 ## Reverse checks - S015 · 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. - S014 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL complete term j=5 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL index times step j=5 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL complete term j=4 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL index times step j=4 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL complete term j=3 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL index times step j=3 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL complete term j=2 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL index times step j=2 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL complete term j=1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL index times step j=1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL complete term j=0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL index times step j=0 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 [9, 6, 3, 0, -3, -6] 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: 7196f49c3641f799035b253e6c165ce5d13930aee207c979c1ecf7bc92663729 - candidate record SHA-256: 15bbc99b80512a14642ea51027fc611662e9725ae179eee8d5dc95903899f9d5 - 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-prefix-0022.json)