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Question 35168: 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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R0069-35168 · q-20260930-r05-sequence-recursive-term-0018

Original question and givens

The real sequence is indexed from 0 with a_0=-5 and recurrence a_(j+1)=a_j+(0) for every j≥0. Find the term at n=12.

  • G001 — The real sequence is indexed from 0 with a_0=-5 and recurrence a_(j+1)=a_j+(0) for every j≥0. Find the term at n=12.

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 = [-5, 0, 12, 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(-5, 0)

After

Sequence ORIGINAL recurrence update j=0 = -5

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, -5, 0]

After

derived = [1, -5]

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(-5, 0)

After

Sequence ORIGINAL recurrence update j=1 = -5

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, -5, 0]

After

derived = [2, -5]

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(-5, 0)

After

Sequence ORIGINAL recurrence update j=2 = -5

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, -5, 0]

After

derived = [3, -5]

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(-5, 0)

After

Sequence ORIGINAL recurrence update j=3 = -5

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, -5, 0]

After

derived = [4, -5]

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(-5, 0)

After

Sequence ORIGINAL recurrence update j=4 = -5

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, -5, 0]

After

derived = [5, -5]

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(-5, 0)

After

Sequence ORIGINAL recurrence update j=5 = -5

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, -5, 0]

After

derived = [6, -5]

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(-5, 0)

After

Sequence ORIGINAL recurrence update j=6 = -5

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, -5, 0]

After

derived = [7, -5]

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(-5, 0)

After

Sequence ORIGINAL recurrence update j=7 = -5

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, -5, 0]

After

derived = [8, -5]

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(-5, 0)

After

Sequence ORIGINAL recurrence update j=8 = -5

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, -5, 0]

After

derived = [9, -5]

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(-5, 0)

After

Sequence ORIGINAL recurrence update j=9 = -5

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, -5, 0]

After

derived = [10, -5]

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(-5, 0)

After

Sequence ORIGINAL recurrence update j=10 = -5

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, -5, 0]

After

derived = [11, -5]

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

S025 · calculation

Before

Sequence ORIGINAL recurrence update j=11 := sum(-5, 0)

After

Sequence ORIGINAL recurrence update j=11 = -5

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, S024

S026 · implies

Before

source roles = [11, -5, 0]

After

derived = [12, -5]

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, S025

Reverse checks

  • S026 · 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.
  • S025 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=11 with exact rational arithmetic, not rounded decimals.
  • 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

-5

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: daa95d78764a073eb8dde66738b4de712f8753897848669b6448bbe4b5f33084
  • candidate record SHA-256: b175c842441ed9970e9094e0925ffb5af43ffbd9acfb9051863e1d05579933d2
  • semantic review receipt SHA-256: ccf93c72638d6abf6a412514da5838d55ecb09b71d0e4fdec2b559dc9fdee14b
  • question-first witness SHA-256: 11af9a338b5a610342b2cfb2ee295683131b3139ac402dd19b1ba1725976d518
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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