第 28173 題：複合梯形法

原題與已知條件

嚴格遞增partition、n+1 nodes與逐subinterval寬度。操作=trap-error-boundary；明示資料={"mode":"sample_table_scope","model":{"nodes":[0,1,25],"values":[22,23,25]}}；請依 exact_ordered_tuple_declared_by_interface 回答。

• G001 — 嚴格遞增partition、n+1 nodes與逐subinterval寬度。操作=trap-error-boundary；明示資料={"mode":"sample_table_scope","model":{"nodes":[0,1,25],"values":[22,23,25]}}；請依 exact_ordered_tuple_declared_by_interface 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-trapezoidal-quadrature

逐步推導

source roles = [qf-mod50-trap-error-boundary, trap-error-boundary]

derived = [[mode, nodes, endpoint_values, trapezoid_value, sum_determined, unique_integrand_or_integral_claim, boundary]]

Trapezoid boundary ORIGINAL ORIGINAL width0 := sub(1, 0)

Trapezoid boundary ORIGINAL ORIGINAL width0 = 1

Trapezoid boundary ORIGINAL ORIGINAL width1 := sub(25, 1)

Trapezoid boundary ORIGINAL ORIGINAL width1 = 24

Trapezoid boundary ORIGINAL ORIGINAL width over2 interval0 := div(1, 2)

Trapezoid boundary ORIGINAL ORIGINAL width over2 interval0 = 1/2

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval0 := sum(22, 23)

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval0 = 45

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval0 := mul(1/2, 45)

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval0 = 45/2

source roles = [Trapezoid boundary ORIGINAL, 0, 0, 1, 22, 23]

derived = [1, 1/2, 45, 45/2]

Trapezoid boundary ORIGINAL ORIGINAL width over2 interval1 := div(24, 2)

Trapezoid boundary ORIGINAL ORIGINAL width over2 interval1 = 12

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval1 := sum(23, 25)

Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval1 = 48

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval1 := mul(12, 48)

Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval1 = 576

source roles = [Trapezoid boundary ORIGINAL, 1, 1, 25, 23, 25]

derived = [24, 12, 48, 576]

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total0 := sum(0, 45/2)

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total0 = 45/2

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total1 := sum(45/2, 576)

Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total1 = 1197/2

source roles = [Trapezoid boundary ORIGINAL, [0, 1, 25], [22, 23, 25]]

derived = [[0, 1, 25], [22, 23, 25], [1, 24], [45/2, 576], 1197/2]

source roles = [[0, 1, 25], [22, 23, 25]]

derived = [1197/2, 是, 否]

逐步反向核對

• S015 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。
• S014 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。
• S013 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL trapezoid running total0，不以小數近似替代。
• S011 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。
• S010 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL width over2 interval1，不以小數近似替代。
• S007 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。
• S006 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL individual trapezoid interval0，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL endpoint sum interval0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL width over2 interval0，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL width1，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Trapezoid boundary ORIGINAL ORIGINAL width0，不以小數近似替代。
• S001 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。

結論

(sample_table_scope, [0, 1, 25], [22, 23, 25], 1197/2, 是, 否, endpoint_table_determines_this_quadrature_sum_not_a_unique_integrand_or_integral)

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。