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Question 35138: Finite Prefix of an Explicit Sequence

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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R0069-35138 · q-20260930-r05-sequence-prefix-0013

Original question and givens

The real sequence is indexed from 0 with a_n=(0)+(2)n. List the prefix [a_0,…,a_(L-1)] of length L=5; use [] when L=0.

  • G001 — The real sequence is indexed from 0 with a_n=(0)+(2)n. List the prefix [a_0,…,a_(L-1)] of length L=5; use [] when L=0.

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-prefix, [L, d, operation, u]]

After

derived = [prefix, 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 = [0, 2, 5, prefix]

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 prefix ORIGINAL index times step j=0 := mul(0, 2)

After

Sequence prefix ORIGINAL index times step j=0 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Sequence prefix ORIGINAL complete term j=0 := sum(0, 0)

After

Sequence prefix ORIGINAL complete term j=0 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Sequence prefix ORIGINAL index times step j=1 := mul(1, 2)

After

Sequence prefix ORIGINAL index times step j=1 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Sequence prefix ORIGINAL complete term j=1 := sum(0, 2)

After

Sequence prefix ORIGINAL complete term j=1 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Sequence prefix ORIGINAL index times step j=2 := mul(2, 2)

After

Sequence prefix ORIGINAL index times step j=2 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Sequence prefix ORIGINAL complete term j=2 := sum(0, 4)

After

Sequence prefix ORIGINAL complete term j=2 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Sequence prefix ORIGINAL index times step j=3 := mul(3, 2)

After

Sequence prefix ORIGINAL index times step j=3 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Sequence prefix ORIGINAL complete term j=3 := sum(0, 6)

After

Sequence prefix ORIGINAL complete term j=3 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Sequence prefix ORIGINAL index times step j=4 := mul(4, 2)

After

Sequence prefix ORIGINAL index times step j=4 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Sequence prefix ORIGINAL complete term j=4 := sum(0, 8)

After

Sequence prefix ORIGINAL complete term j=4 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

source roles = [0, 2, 5]

After

derived = [[0, 2, 4, 6, 8], [0, 2, 4, 6, 8]]

Conditions: none

Take EXACTLY L terms in index order from a0 through a_(L−1),NOT through aL.L=0 displays[].The GIVEN formula determines these terms;the prefix alone need not determine an unknown rule.

Step identifiers and dependencies

Rule: Sequence_PREFIX_EXACT_length_L_indices0_to_Lminus1 · Uses: G001, S012

Reverse checks

  • S013 · 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.
  • S012 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL complete term j=4 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL index times step j=4 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL complete term j=3 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL index times step j=3 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL complete term j=2 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL index times step j=2 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL complete term j=1 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL index times step j=1 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL complete term j=0 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Sequence prefix ORIGINAL index times step 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

[0, 2, 4, 6, 8]

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: aba1b76340649377d3a704504dceeec35b7eb318185c3b833659f5daaae8dc0a
  • candidate record SHA-256: 38f944c036853fb1ff625730efc5ac99d54f97e0a7fc3c508a9c02b3e06db071
  • semantic review receipt SHA-256: ccf93c72638d6abf6a412514da5838d55ecb09b71d0e4fdec2b559dc9fdee14b
  • question-first witness SHA-256: 11af9a338b5a610342b2cfb2ee295683131b3139ac402dd19b1ba1725976d518
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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