Use the entire arithmetic source a_n=7+(7)n for n≥0 and index map k(j)=2+2j, with nonnegative start and positive index step. Evaluate j=8. Give (j, source_index, subsequence_term, certificate).
G001 — Use the entire arithmetic source a_n=7+(7)n for n≥0 and index map k(j)=2+2j, with nonnegative start and positive index step. Evaluate j=8. Give (j, source_index, subsequence_term, certificate).
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.
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope.
original subsequence index k:ORIGINAL nonnegative start=2;positive integer step=2;F(q)=start+step*q for EVERY natural q>=0
After
F(q)>=start>=0 and for ANY q<r,F(r)-F(q)=(2)*(r-q)>0;strictly increasing on ALL naturals,not just the evaluated prefix
Conditions: none
A positive integer step preserves the order of every pair of natural indices,not merely injectivity or an arbitrary reordering.This universal mathematical law is not truncated by an evaluated-output safety range and says nothing about monotonicity of source values.
ORIGINAL arithmetic source a_n=7+(7)n;increasing selected index k(8)=18
After
a_at_selected_index=133;ORIGINAL index order retained even if source step=7 is negative or0;no value sorting
Conditions: none
Compute the selected index and substitute it into the original source formula.Source steps may be negative or0:a subsequence preserves index order,not increasing values.Sorting the resulting values would not produce this stated subsequence.
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S006 · premise_and_result_audit — Recompute subsequence value of ORIGINAL source at selected index with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute subsequence ORIGINAL source step times selected index with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute subsequence selected ORIGINAL source index with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute subsequence selected index step times j 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
(8, 18, 133, a_of_k_j_preserves_original_order)
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.