第 20412 題：子數列

原題與已知條件

自然數索引與symbolic嚴格遞增證書。操作=subseq-arithmetic-term；資料={"source_start":12,"source_step":5,"index_start":3,"index_step":2,"j":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 自然數索引與symbolic嚴格遞增證書。操作=subseq-arithmetic-term；資料={"source_start":12,"source_step":5,"index_start":3,"index_step":2,"j":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-subsequence

逐步推導

source operation=subseq-arithmetic-term;original snapshot and mathematics retained

explicit ordered source output fields=[j, source_index, subsequence_term, certificate]

original subsequence index k:ORIGINAL nonnegative start=3;positive integer step=2;F(q)=start+step*q for EVERY natural q>=0

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

subsequence selected index step times j := mul(2, 13)

subsequence selected index step times j = 26

subsequence selected ORIGINAL source index := sum(3, 26)

subsequence selected ORIGINAL source index = 29

subsequence ORIGINAL source step times selected index := mul(5, 29)

subsequence ORIGINAL source step times selected index = 145

subsequence value of ORIGINAL source at selected index := sum(12, 145)

subsequence value of ORIGINAL source at selected index = 157

ORIGINAL arithmetic source a_n=12+(5)n;increasing selected index k(13)=29

a_at_selected_index=157;ORIGINAL index order retained even if source step=5 is negative or0;no value sorting

逐步反向核對

• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 以精確有理數重算 subsequence value of ORIGINAL source at selected index，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 subsequence ORIGINAL source step times selected index，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 subsequence selected ORIGINAL source index，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 subsequence selected index step times j，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(13, 29, 157, a_of_k_j_preserves_original_order)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。