Question 28182: n Subintervals Require n+1 Aligned Strictly Increasing Nodes

Original question and givens

Use ordered nodes=[0, 1, 9] and aligned endpoint values=[6, 7, 9]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).

• G001 — Use ordered nodes=[0, 1, 9] and aligned endpoint values=[6, 7, 9]. Give (nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention).

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-trapezoidal-quadrature

Step-by-step derivation

source roles = [qf-mod50-trap-partition-qualification, trap-partition-qualification]

derived = [[nodes, endpoint_values, subinterval_count, node_count, individual_widths, partition_qualified, node_count_convention]]

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

Trapezoid ORIGINAL ORIGINAL width0 = 1

Trapezoid ORIGINAL ORIGINAL width1 := sub(9, 1)

Trapezoid ORIGINAL ORIGINAL width1 = 8

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

Trapezoid ORIGINAL ORIGINAL width over2 interval0 = 1/2

Trapezoid ORIGINAL ORIGINAL endpoint sum interval0 := sum(6, 7)

Trapezoid ORIGINAL ORIGINAL endpoint sum interval0 = 13

Trapezoid ORIGINAL ORIGINAL individual trapezoid interval0 := mul(1/2, 13)

Trapezoid ORIGINAL ORIGINAL individual trapezoid interval0 = 13/2

source roles = [Trapezoid ORIGINAL, 0, 0, 1, 6, 7]

derived = [1, 1/2, 13, 13/2]

Trapezoid ORIGINAL ORIGINAL width over2 interval1 := div(8, 2)

Trapezoid ORIGINAL ORIGINAL width over2 interval1 = 4

Trapezoid ORIGINAL ORIGINAL endpoint sum interval1 := sum(7, 9)

Trapezoid ORIGINAL ORIGINAL endpoint sum interval1 = 16

Trapezoid ORIGINAL ORIGINAL individual trapezoid interval1 := mul(4, 16)

Trapezoid ORIGINAL ORIGINAL individual trapezoid interval1 = 64

source roles = [Trapezoid ORIGINAL, 1, 1, 9, 7, 9]

derived = [8, 4, 16, 64]

Trapezoid ORIGINAL ORIGINAL trapezoid running total0 := sum(0, 13/2)

Trapezoid ORIGINAL ORIGINAL trapezoid running total0 = 13/2

Trapezoid ORIGINAL ORIGINAL trapezoid running total1 := sum(13/2, 64)

Trapezoid ORIGINAL ORIGINAL trapezoid running total1 = 141/2

source roles = [Trapezoid ORIGINAL, [0, 1, 9], [6, 7, 9]]

derived = [[0, 1, 9], [6, 7, 9], [1, 8], [13/2, 64], 141/2]

Reverse checks

• S014 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S013 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL trapezoid running total1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL trapezoid running total0 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S010 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL individual trapezoid interval1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL endpoint sum interval1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL width over2 interval1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S006 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL individual trapezoid interval0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL endpoint sum interval0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL width over2 interval0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL width1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL width0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.

Result

([0, 1, 9], [6, 7, 9], 2, 3, [1, 8], Yes, n_subintervals_have_n_plus_one_nodes)

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.