# Question 20409: A Subsequence Term Uses the Source at Its Increasing Selected Index [Return to catalogue page 103](https://ocme.evemisslab.com/en/questions/page/103/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261107-r07-subsequence-foundations/#q-20261014-advance-20261107-r07-subsequence-0034) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0039-20409 · q-20261014-advance-20261107-r07-subsequence-0034 ## Original question and givens Use the entire arithmetic source a_n=9+(2)n for n≥0 and index map k(j)=0+4j, with nonnegative start and positive index step. Evaluate j=10. Give (j, source_index, subsequence_term, certificate). - G001 — Use the entire arithmetic source a_n=9+(2)n for n≥0 and index map k(j)=0+4j, with nonnegative start and positive index step. Evaluate j=10. 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. - [mko-subsequence](https://ocme.evemisslab.com/en/mko/mko-subsequence/) ## Step-by-step derivation S001 · impliesBefore source operation=subseq-arithmetic-term;original snapshot and mathematics retained After explicit ordered source output fields=[j, source_index, subsequence_term, certificate] Conditions: none 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. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore original subsequence index k:ORIGINAL nonnegative start=0;positive integer step=4;F(q)=start+step*q for EVERY natural q>=0 After F(q)>=start>=0 and for ANY q0;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. Step identifiers and dependenciesRule: subseq_affine_all_natural_index_order_not_value_monotonicity · Uses: G001, S001 S003 · calculationBefore subsequence selected index step times j := mul(4, 10) After subsequence selected index step times j = 40 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore subsequence selected ORIGINAL source index := sum(0, 40) After subsequence selected ORIGINAL source index = 40 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore subsequence ORIGINAL source step times selected index := mul(2, 40) After subsequence ORIGINAL source step times selected index = 80 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore subsequence value of ORIGINAL source at selected index := sum(9, 80) After subsequence value of ORIGINAL source at selected index = 89 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore ORIGINAL arithmetic source a_n=9+(2)n;increasing selected index k(10)=40 After a_at_selected_index=89;ORIGINAL index order retained even if source step=2 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. Step identifiers and dependenciesRule: subseq_selected_source_order_not_reordered_values · Uses: G001, S006 ## Reverse checks - 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 (10, 40, 89, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 2f84d2b0447240cd8687115e9a10d33c4eed24fb6cee71d5377a12c7bab50c9d - candidate record SHA-256: ef923f8349ca3e332d677f9a242cb7926f4f05d27e6267efa9649bc2154baa1b - semantic review receipt SHA-256: 6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034 - question-first witness SHA-256: cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261107-r07-subsequence-0034.json)