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Question 5412: 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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R0009-05412 · q-20261008-advance-20261012-r01-mathematical-induction-0037

Original question and givens

N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-1, a(k+1)=a(k)+(-1), through=4. 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)=-1, a(k+1)=a(k)+(-1), through=4. 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)=-1;through=4

After

[-1]

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(-1, -1)

After

recurrence next value1 = -2

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)=-1;d=-1

After

a(1)=-2

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(-2, -1)

After

recurrence next value2 = -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, S004

S006 · implies

Before

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

After

a(2)=-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, S005

S007 · calculation

Before

recurrence next value3 := sum(-3, -1)

After

recurrence next value3 = -4

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)=-3;d=-1

After

a(3)=-4

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(-4, -1)

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)=-4;d=-1

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

Reverse checks

  • 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

[-1, -2, -3, -4, -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: 5eba17c5c818554032de68e6aa2c4fef2cc7d0bb5d406fcefdf45fe743ea248a
  • candidate record SHA-256: 830496b4edb0495952d6ec60333d5e764a91eac43ef390d537adf83e5772feb0
  • semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
  • question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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