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Question 5416: Inclusive Finite Recurrence Prefix

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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R0009-05416 · q-20261008-advance-20261012-r01-mathematical-induction-0041

Original question and givens

N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=3, a(k+1)=a(k)+(-2), through=8. List a(0) through a(through), inclusive.

  • G001 — N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=3, a(k+1)=a(k)+(-2), through=8. List a(0) through a(through), inclusive.

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

natural indices start at0;finite induction protocol

After

index bookkeeping and finite values are not proofs of all P(n)

Conditions: none

Natural indices do not require nonnegative sequence values.Checking a finite range or a successor index does not supply proposition truth or an induction proof over all naturals.

Step identifiers and dependencies

Rule: finite_induction_claim_boundary · Uses: G001

S002 · implies

Before

a(0)=3;through=8

After

[3]

Conditions: none

Start with the supplied initial value;through=0 retains only it.

Step identifiers and dependencies

Rule: recurrence_prefix_base · Uses: G001, S001

S003 · calculation

Before

recurrence next value1 := sum(3, -2)

After

recurrence next value1 = 1

Conditions: none

Add the declared d to the previous value;negative increments or terms are allowed.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

a(0)=3;d=-2

After

a(1)=1

Conditions: none

Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.

Step identifiers and dependencies

Rule: finite_recurrence_transition · Uses: G001, S003

S005 · calculation

Before

recurrence next value2 := sum(1, -2)

After

recurrence next value2 = -1

Conditions: none

Add the declared d to the previous value;negative increments or terms are allowed.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

a(1)=1;d=-2

After

a(2)=-1

Conditions: none

Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.

Step identifiers and dependencies

Rule: finite_recurrence_transition · Uses: G001, S005

S007 · calculation

Before

recurrence next value3 := sum(-1, -2)

After

recurrence next value3 = -3

Conditions: none

Add the declared d to the previous value;negative increments or terms are allowed.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

a(2)=-1;d=-2

After

a(3)=-3

Conditions: none

Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.

Step identifiers and dependencies

Rule: finite_recurrence_transition · Uses: G001, S007

S009 · calculation

Before

recurrence next value4 := sum(-3, -2)

After

recurrence next value4 = -5

Conditions: none

Add the declared d to the previous value;negative increments or terms are allowed.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

a(3)=-3;d=-2

After

a(4)=-5

Conditions: none

Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.

Step identifiers and dependencies

Rule: finite_recurrence_transition · Uses: G001, S009

S011 · calculation

Before

recurrence next value5 := sum(-5, -2)

After

recurrence next value5 = -7

Conditions: none

Add the declared d to the previous value;negative increments or terms are allowed.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

a(4)=-5;d=-2

After

a(5)=-7

Conditions: none

Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.

Step identifiers and dependencies

Rule: finite_recurrence_transition · Uses: G001, S011

S013 · calculation

Before

recurrence next value6 := sum(-7, -2)

After

recurrence next value6 = -9

Conditions: none

Add the declared d to the previous value;negative increments or terms are allowed.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

a(5)=-7;d=-2

After

a(6)=-9

Conditions: none

Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.

Step identifiers and dependencies

Rule: finite_recurrence_transition · Uses: G001, S013

S015 · calculation

Before

recurrence next value7 := sum(-9, -2)

After

recurrence next value7 = -11

Conditions: none

Add the declared d to the previous value;negative increments or terms are allowed.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

a(6)=-9;d=-2

After

a(7)=-11

Conditions: none

Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.

Step identifiers and dependencies

Rule: finite_recurrence_transition · Uses: G001, S015

S017 · calculation

Before

recurrence next value8 := sum(-11, -2)

After

recurrence next value8 = -13

Conditions: none

Add the declared d to the previous value;negative increments or terms are allowed.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

a(7)=-11;d=-2

After

a(8)=-13

Conditions: none

Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.

Step identifiers and dependencies

Rule: finite_recurrence_transition · Uses: G001, S017

Reverse checks

  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Recompute recurrence next value8 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Recompute recurrence next value7 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Recompute recurrence next value6 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Recompute recurrence next value5 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Recompute recurrence next value4 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Recompute recurrence next value3 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Recompute recurrence next value2 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Recompute recurrence next value1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

[3, 1, -1, -3, -5, -7, -9, -11, -13]

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: 832ce347a4430b9c9fd8c6f9bcccd7953782a94a20b466c186093e97395c81be
  • candidate record SHA-256: 3d82dcf5abc65427a46d95e7c6b6bd4fbf9357ee83a8a0b7040b79905f839c05
  • semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
  • question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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