Question 32603: Finite Left-Endpoint Sum of an Affine Function

Original question and givens

On all R, f(x)=(3)x+(-2). For a=1≤b=1 and N=4 uniform cells, compute the finite left-endpoint sum with Δx=(b−a)/N and x_i=a+iΔx for i=0,…,N−1. Give a reduced fraction with positive denominator, retaining /1 for integers.

• G001 — On all R, f(x)=(3)x+(-2). For a=1≤b=1 and N=4 uniform cells, compute the finite left-endpoint sum with Δx=(b−a)/N and x_i=a+iΔx for i=0,…,N−1. Give a reduced fraction with positive denominator, retaining /1 for integers.

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-definite-integral

Step-by-step derivation

source roles = [qf-integral-affine-left-sum, affine_left_sum, reals, []]

derived = [[finite_left_sum_fraction_RETAIN1]]

Left sum ORIGINAL upper minus lower endpoint := sub(1, 1)

Left sum ORIGINAL upper minus lower endpoint = 0

Left sum ORIGINAL width over positive cell count := div(0, 4)

Left sum ORIGINAL width over positive cell count = 0

Left sum ORIGINAL sample0 index times width := mul(0, 0)

Left sum ORIGINAL sample0 index times width = 0

Left sum ORIGINAL sample0 lower plus offset := sum(1, 0)

Left sum ORIGINAL sample0 lower plus offset = 1

Left sum ORIGINAL sample0 slope times x := mul(3, 1)

Left sum ORIGINAL sample0 slope times x = 3

Left sum ORIGINAL sample0 affine value plus c := sum(3, -2)

Left sum ORIGINAL sample0 affine value plus c = 1

Left sum ORIGINAL sample0 value times cell width := mul(1, 0)

Left sum ORIGINAL sample0 value times cell width = 0

Left sum ORIGINAL accumulated term0 := sum(0, 0)

Left sum ORIGINAL accumulated term0 = 0

source roles = [0, 4, 3, -2, 1, 1, 0/1]

derived = [1/1, 1/1, 0/1, 0/1]

Left sum ORIGINAL sample1 index times width := mul(1, 0)

Left sum ORIGINAL sample1 index times width = 0

Left sum ORIGINAL sample1 lower plus offset := sum(1, 0)

Left sum ORIGINAL sample1 lower plus offset = 1

Left sum ORIGINAL sample1 slope times x := mul(3, 1)

Left sum ORIGINAL sample1 slope times x = 3

Left sum ORIGINAL sample1 affine value plus c := sum(3, -2)

Left sum ORIGINAL sample1 affine value plus c = 1

Left sum ORIGINAL sample1 value times cell width := mul(1, 0)

Left sum ORIGINAL sample1 value times cell width = 0

Left sum ORIGINAL accumulated term1 := sum(0, 0)

Left sum ORIGINAL accumulated term1 = 0

source roles = [1, 4, 3, -2, 1, 1, 0/1]

derived = [1/1, 1/1, 0/1, 0/1]

Left sum ORIGINAL sample2 index times width := mul(2, 0)

Left sum ORIGINAL sample2 index times width = 0

Left sum ORIGINAL sample2 lower plus offset := sum(1, 0)

Left sum ORIGINAL sample2 lower plus offset = 1

Left sum ORIGINAL sample2 slope times x := mul(3, 1)

Left sum ORIGINAL sample2 slope times x = 3

Left sum ORIGINAL sample2 affine value plus c := sum(3, -2)

Left sum ORIGINAL sample2 affine value plus c = 1

Left sum ORIGINAL sample2 value times cell width := mul(1, 0)

Left sum ORIGINAL sample2 value times cell width = 0

Left sum ORIGINAL accumulated term2 := sum(0, 0)

Left sum ORIGINAL accumulated term2 = 0

source roles = [2, 4, 3, -2, 1, 1, 0/1]

derived = [1/1, 1/1, 0/1, 0/1]

Left sum ORIGINAL sample3 index times width := mul(3, 0)

Left sum ORIGINAL sample3 index times width = 0

Left sum ORIGINAL sample3 lower plus offset := sum(1, 0)

Left sum ORIGINAL sample3 lower plus offset = 1

Left sum ORIGINAL sample3 slope times x := mul(3, 1)

Left sum ORIGINAL sample3 slope times x = 3

Left sum ORIGINAL sample3 affine value plus c := sum(3, -2)

Left sum ORIGINAL sample3 affine value plus c = 1

Left sum ORIGINAL sample3 value times cell width := mul(1, 0)

Left sum ORIGINAL sample3 value times cell width = 0

Left sum ORIGINAL accumulated term3 := sum(0, 0)

Left sum ORIGINAL accumulated term3 = 0

source roles = [3, 4, 3, -2, 1, 1, 0/1]

derived = [1/1, 1/1, 0/1, 0/1]

source roles = [4, 3, -2, 1, 1]

derived = [0/1, 0/1]

Reverse checks

• S032 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S031 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S030 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term3 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 value times cell width with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 affine value plus c with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 slope times x with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 lower plus offset with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 index times width with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S023 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term2 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 value times cell width with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 affine value plus c with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 slope times x with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 lower plus offset with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 index times width with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S016 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term1 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 value times cell width with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 affine value plus c with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 slope times x with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 lower plus offset with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 index times width with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S009 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 value times cell width with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 affine value plus c with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 slope times x with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 lower plus offset with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 index times width with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Left sum ORIGINAL width over positive cell count with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Left sum ORIGINAL upper minus lower endpoint with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

0/1

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.