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R0069-35145 · q-20260930-r05-sequence-prefix-0020
The real sequence is indexed from 0 with a_n=(7)+(2)n. List the prefix [a_0,…,a_(L-1)] of length L=4; use [] when L=0.
G001 — The real sequence is indexed from 0 with a_n=(7)+(2)n. List the prefix [a_0,…,a_(L-1)] of length L=4; use [] when L=0.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.
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.
Rule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001
Before
source roles = [7, 2, 4, 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.
Rule: Sequence_ALL_RAW_initial_step_and_count_EVEN_empty_prefix · Uses: G001, S001
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Sequence prefix ORIGINAL complete term j=0 := sum(7, 0)
After
Sequence prefix ORIGINAL complete term j=0 = 7
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Sequence prefix ORIGINAL complete term j=1 := sum(7, 2)
After
Sequence prefix ORIGINAL complete term j=1 = 9
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
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.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Sequence prefix ORIGINAL complete term j=2 := sum(7, 4)
After
Sequence prefix ORIGINAL complete term j=2 = 11
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.
Rule: exact_rational_arithmetic · Uses: G001, S007
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.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
Sequence prefix ORIGINAL complete term j=3 := sum(7, 6)
After
Sequence prefix ORIGINAL complete term j=3 = 13
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.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
source roles = [7, 2, 4]
After
derived = [[7, 9, 11, 13], [7, 9, 11, 13]]
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.
Rule: Sequence_PREFIX_EXACT_length_L_indices0_to_Lminus1 · Uses: G001, S010
S011 · 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.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.[7, 9, 11, 13]
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 974c9b329dfacf2073b5b00cab59e3cb5d27b00019df3cdb3a494398aff9a69ecandidate record SHA-256: 432797189fa7b37b2da989d5c17e3038d778d2e2b700b52b1c856fc69a584a46semantic review receipt SHA-256: ccf93c72638d6abf6a412514da5838d55ecb09b71d0e4fdec2b559dc9fdee14bquestion-first witness SHA-256: 11af9a338b5a610342b2cfb2ee295683131b3139ac402dd19b1ba1725976d518Lean: NOT_RUN · human review: NOT_PERFORMED