# Question 35159: Term of a Recurrence with an Initial Value [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-recursive-term-0009) 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. R0069-35159 · q-20260930-r05-sequence-recursive-term-0009 ## Original question and givens The real sequence is indexed from 0 with a_0=4 and recurrence a_(j+1)=a_j+(-2) for every j≥0. Find the term at n=11. - G001 — The real sequence is indexed from 0 with a_0=4 and recurrence a_(j+1)=a_j+(-2) for every j≥0. Find the term at n=11. ## 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 = [4, -2, 11, 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 S003 · calculationBefore Sequence ORIGINAL recurrence update j=0 := sum(4, -2) After Sequence ORIGINAL recurrence update j=0 = 2 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 · impliesBefore source roles = [0, 4, -2] After derived = [1, 2] Conditions: none At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates. Step identifiers and dependenciesRule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S003 S005 · calculationBefore Sequence ORIGINAL recurrence update j=1 := sum(2, -2) After Sequence ORIGINAL recurrence update j=1 = 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, S004 S006 · impliesBefore source roles = [1, 2, -2] After derived = [2, 0] Conditions: none At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates. Step identifiers and dependenciesRule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S005 S007 · calculationBefore Sequence ORIGINAL recurrence update j=2 := sum(0, -2) After Sequence ORIGINAL recurrence update j=2 = -2 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 · impliesBefore source roles = [2, 0, -2] After derived = [3, -2] Conditions: none At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates. Step identifiers and dependenciesRule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S007 S009 · calculationBefore Sequence ORIGINAL recurrence update j=3 := sum(-2, -2) After Sequence ORIGINAL recurrence update j=3 = -4 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 · impliesBefore source roles = [3, -2, -2] After derived = [4, -4] Conditions: none At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates. Step identifiers and dependenciesRule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S009 S011 · calculationBefore Sequence ORIGINAL recurrence update j=4 := sum(-4, -2) After Sequence ORIGINAL recurrence update j=4 = -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, S010 S012 · impliesBefore source roles = [4, -4, -2] After derived = [5, -6] Conditions: none At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates. Step identifiers and dependenciesRule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S011 S013 · calculationBefore Sequence ORIGINAL recurrence update j=5 := sum(-6, -2) After Sequence ORIGINAL recurrence update j=5 = -8 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 · impliesBefore source roles = [5, -6, -2] After derived = [6, -8] Conditions: none At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates. Step identifiers and dependenciesRule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S013 S015 · calculationBefore Sequence ORIGINAL recurrence update j=6 := sum(-8, -2) After Sequence ORIGINAL recurrence update j=6 = -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, S014 S016 · impliesBefore source roles = [6, -8, -2] After derived = [7, -10] Conditions: none At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates. Step identifiers and dependenciesRule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S015 S017 · calculationBefore Sequence ORIGINAL recurrence update j=7 := sum(-10, -2) After Sequence ORIGINAL recurrence update j=7 = -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, S016 S018 · impliesBefore source roles = [7, -10, -2] After derived = [8, -12] Conditions: none At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates. Step identifiers and dependenciesRule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S017 S019 · calculationBefore Sequence ORIGINAL recurrence update j=8 := sum(-12, -2) After Sequence ORIGINAL recurrence update j=8 = -14 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, S018 S020 · impliesBefore source roles = [8, -12, -2] After derived = [9, -14] Conditions: none At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates. Step identifiers and dependenciesRule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S019 S021 · calculationBefore Sequence ORIGINAL recurrence update j=9 := sum(-14, -2) After Sequence ORIGINAL recurrence update j=9 = -16 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, S020 S022 · impliesBefore source roles = [9, -14, -2] After derived = [10, -16] Conditions: none At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates. Step identifiers and dependenciesRule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S021 S023 · calculationBefore Sequence ORIGINAL recurrence update j=10 := sum(-16, -2) After Sequence ORIGINAL recurrence update j=10 = -18 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, S022 S024 · impliesBefore source roles = [10, -16, -2] After derived = [11, -18] Conditions: none At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates. Step identifiers and dependenciesRule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S023 ## Reverse checks - S024 · 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. - S023 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=10 with exact rational arithmetic, not rounded decimals. - S022 · 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. - S021 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=9 with exact rational arithmetic, not rounded decimals. - S020 · 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. - S019 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=8 with exact rational arithmetic, not rounded decimals. - S018 · 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. - S017 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=7 with exact rational arithmetic, not rounded decimals. - S016 · 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. - S015 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=6 with exact rational arithmetic, not rounded decimals. - S014 · 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. - S013 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=5 with exact rational arithmetic, not rounded decimals. - S012 · 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. - S011 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=4 with exact rational arithmetic, not rounded decimals. - S010 · 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. - S009 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=3 with exact rational arithmetic, not rounded decimals. - S008 · 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. - S007 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=2 with exact rational arithmetic, not rounded decimals. - S006 · 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. - S005 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=1 with exact rational arithmetic, not rounded decimals. - S004 · 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. - S003 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update 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 -18 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: 578b4b2fab224e3c6aadb7972bc6e285a4ad7e7d9f97c89caa40fe1555a0677c - candidate record SHA-256: c104960773b4e30fd6d09a15b7a42b17c674637ac4910262855834f1c10e5388 - 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-0009.json)