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Question 35172: Term of a Recurrence with an Initial Value

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0069-35172 · q-20260930-r05-sequence-recursive-term-0022

Original question and givens

The real sequence is indexed from 0 with a_0=-9 and recurrence a_(j+1)=a_j+(-3) for every j≥0. Find the term at n=11.

  • G001 — The real sequence is indexed from 0 with a_0=-9 and recurrence a_(j+1)=a_j+(-3) 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001

S002 · implies

Before

source roles = [-9, -3, 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 dependencies

Rule: Sequence_ALL_RAW_initial_step_and_count_EVEN_empty_prefix · Uses: G001, S001

S003 · calculation

Before

Sequence ORIGINAL recurrence update j=0 := sum(-9, -3)

After

Sequence ORIGINAL recurrence update j=0 = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [0, -9, -3]

After

derived = [1, -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 dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S003

S005 · calculation

Before

Sequence ORIGINAL recurrence update j=1 := sum(-12, -3)

After

Sequence ORIGINAL recurrence update j=1 = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [1, -12, -3]

After

derived = [2, -15]

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 dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S005

S007 · calculation

Before

Sequence ORIGINAL recurrence update j=2 := sum(-15, -3)

After

Sequence ORIGINAL recurrence update j=2 = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [2, -15, -3]

After

derived = [3, -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 dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S007

S009 · calculation

Before

Sequence ORIGINAL recurrence update j=3 := sum(-18, -3)

After

Sequence ORIGINAL recurrence update j=3 = -21

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [3, -18, -3]

After

derived = [4, -21]

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 dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S009

S011 · calculation

Before

Sequence ORIGINAL recurrence update j=4 := sum(-21, -3)

After

Sequence ORIGINAL recurrence update j=4 = -24

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [4, -21, -3]

After

derived = [5, -24]

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 dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S011

S013 · calculation

Before

Sequence ORIGINAL recurrence update j=5 := sum(-24, -3)

After

Sequence ORIGINAL recurrence update j=5 = -27

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [5, -24, -3]

After

derived = [6, -27]

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 dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S013

S015 · calculation

Before

Sequence ORIGINAL recurrence update j=6 := sum(-27, -3)

After

Sequence ORIGINAL recurrence update j=6 = -30

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

source roles = [6, -27, -3]

After

derived = [7, -30]

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 dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S015

S017 · calculation

Before

Sequence ORIGINAL recurrence update j=7 := sum(-30, -3)

After

Sequence ORIGINAL recurrence update j=7 = -33

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

source roles = [7, -30, -3]

After

derived = [8, -33]

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 dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S017

S019 · calculation

Before

Sequence ORIGINAL recurrence update j=8 := sum(-33, -3)

After

Sequence ORIGINAL recurrence update j=8 = -36

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

source roles = [8, -33, -3]

After

derived = [9, -36]

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 dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S019

S021 · calculation

Before

Sequence ORIGINAL recurrence update j=9 := sum(-36, -3)

After

Sequence ORIGINAL recurrence update j=9 = -39

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

source roles = [9, -36, -3]

After

derived = [10, -39]

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 dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S021

S023 · calculation

Before

Sequence ORIGINAL recurrence update j=10 := sum(-39, -3)

After

Sequence ORIGINAL recurrence update j=10 = -42

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

source roles = [10, -39, -3]

After

derived = [11, -42]

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 dependencies

Rule: 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

-42

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: e865693cb2a97dad812d5dc3121b4de67b4a5a15a8e89c790d7369cb67fc497e
  • candidate record SHA-256: 33cf326309e42d76666e53c2428e0e245c5f2dd4d6ab26e9a972d99ce745a74c
  • semantic review receipt SHA-256: ccf93c72638d6abf6a412514da5838d55ecb09b71d0e4fdec2b559dc9fdee14b
  • question-first witness SHA-256: 11af9a338b5a610342b2cfb2ee295683131b3139ac402dd19b1ba1725976d518
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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