Question 32606: Finite Left-Endpoint Sum of an Affine Function

Original question and givens

On all R, f(x)=(-6)x+(-7). For a=-1≤b=0 and N=6 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)=(-6)x+(-7). For a=-1≤b=0 and N=6 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(0, -1)

Left sum ORIGINAL upper minus lower endpoint = 1

Left sum ORIGINAL width over positive cell count := div(1, 6)

Left sum ORIGINAL width over positive cell count = 1/6

Left sum ORIGINAL sample0 index times width := mul(0, 1/6)

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(-6, -1)

Left sum ORIGINAL sample0 slope times x = 6

Left sum ORIGINAL sample0 affine value plus c := sum(6, -7)

Left sum ORIGINAL sample0 affine value plus c = -1

Left sum ORIGINAL sample0 value times cell width := mul(-1, 1/6)

Left sum ORIGINAL sample0 value times cell width = -1/6

Left sum ORIGINAL accumulated term0 := sum(0, -1/6)

Left sum ORIGINAL accumulated term0 = -1/6

source roles = [0, 6, -6, -7, -1, 0, 1/6]

derived = [-1/1, -1/1, -1/6, -1/6]

Left sum ORIGINAL sample1 index times width := mul(1, 1/6)

Left sum ORIGINAL sample1 index times width = 1/6

Left sum ORIGINAL sample1 lower plus offset := sum(-1, 1/6)

Left sum ORIGINAL sample1 lower plus offset = -5/6

Left sum ORIGINAL sample1 slope times x := mul(-6, -5/6)

Left sum ORIGINAL sample1 slope times x = 5

Left sum ORIGINAL sample1 affine value plus c := sum(5, -7)

Left sum ORIGINAL sample1 affine value plus c = -2

Left sum ORIGINAL sample1 value times cell width := mul(-2, 1/6)

Left sum ORIGINAL sample1 value times cell width = -1/3

Left sum ORIGINAL accumulated term1 := sum(-1/6, -1/3)

Left sum ORIGINAL accumulated term1 = -1/2

source roles = [1, 6, -6, -7, -1, 0, 1/6]

derived = [-5/6, -2/1, -1/3, -1/2]

Left sum ORIGINAL sample2 index times width := mul(2, 1/6)

Left sum ORIGINAL sample2 index times width = 1/3

Left sum ORIGINAL sample2 lower plus offset := sum(-1, 1/3)

Left sum ORIGINAL sample2 lower plus offset = -2/3

Left sum ORIGINAL sample2 slope times x := mul(-6, -2/3)

Left sum ORIGINAL sample2 slope times x = 4

Left sum ORIGINAL sample2 affine value plus c := sum(4, -7)

Left sum ORIGINAL sample2 affine value plus c = -3

Left sum ORIGINAL sample2 value times cell width := mul(-3, 1/6)

Left sum ORIGINAL sample2 value times cell width = -1/2

Left sum ORIGINAL accumulated term2 := sum(-1/2, -1/2)

Left sum ORIGINAL accumulated term2 = -1

source roles = [2, 6, -6, -7, -1, 0, 1/6]

derived = [-2/3, -3/1, -1/2, -1/1]

Left sum ORIGINAL sample3 index times width := mul(3, 1/6)

Left sum ORIGINAL sample3 index times width = 1/2

Left sum ORIGINAL sample3 lower plus offset := sum(-1, 1/2)

Left sum ORIGINAL sample3 lower plus offset = -1/2

Left sum ORIGINAL sample3 slope times x := mul(-6, -1/2)

Left sum ORIGINAL sample3 slope times x = 3

Left sum ORIGINAL sample3 affine value plus c := sum(3, -7)

Left sum ORIGINAL sample3 affine value plus c = -4

Left sum ORIGINAL sample3 value times cell width := mul(-4, 1/6)

Left sum ORIGINAL sample3 value times cell width = -2/3

Left sum ORIGINAL accumulated term3 := sum(-1, -2/3)

Left sum ORIGINAL accumulated term3 = -5/3

source roles = [3, 6, -6, -7, -1, 0, 1/6]

derived = [-1/2, -4/1, -2/3, -5/3]

Left sum ORIGINAL sample4 index times width := mul(4, 1/6)

Left sum ORIGINAL sample4 index times width = 2/3

Left sum ORIGINAL sample4 lower plus offset := sum(-1, 2/3)

Left sum ORIGINAL sample4 lower plus offset = -1/3

Left sum ORIGINAL sample4 slope times x := mul(-6, -1/3)

Left sum ORIGINAL sample4 slope times x = 2

Left sum ORIGINAL sample4 affine value plus c := sum(2, -7)

Left sum ORIGINAL sample4 affine value plus c = -5

Left sum ORIGINAL sample4 value times cell width := mul(-5, 1/6)

Left sum ORIGINAL sample4 value times cell width = -5/6

Left sum ORIGINAL accumulated term4 := sum(-5/3, -5/6)

Left sum ORIGINAL accumulated term4 = -5/2

source roles = [4, 6, -6, -7, -1, 0, 1/6]

derived = [-1/3, -5/1, -5/6, -5/2]

Left sum ORIGINAL sample5 index times width := mul(5, 1/6)

Left sum ORIGINAL sample5 index times width = 5/6

Left sum ORIGINAL sample5 lower plus offset := sum(-1, 5/6)

Left sum ORIGINAL sample5 lower plus offset = -1/6

Left sum ORIGINAL sample5 slope times x := mul(-6, -1/6)

Left sum ORIGINAL sample5 slope times x = 1

Left sum ORIGINAL sample5 affine value plus c := sum(1, -7)

Left sum ORIGINAL sample5 affine value plus c = -6

Left sum ORIGINAL sample5 value times cell width := mul(-6, 1/6)

Left sum ORIGINAL sample5 value times cell width = -1

Left sum ORIGINAL accumulated term5 := sum(-5/2, -1)

Left sum ORIGINAL accumulated term5 = -7/2

source roles = [5, 6, -6, -7, -1, 0, 1/6]

derived = [-1/6, -6/1, -1/1, -7/2]

source roles = [6, -6, -7, -1, 0]

derived = [1/6, -7/2]

Reverse checks

• S046 · 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.
• S045 · 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.
• S044 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term5 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 value times cell width with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 affine value plus c with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 slope times x with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 lower plus offset with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 index times width with exact rational arithmetic, not rounded decimals.
• S038 · 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.
• S037 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term4 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 value times cell width with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 affine value plus c with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 slope times x with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 lower plus offset with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 index times width with exact rational arithmetic, not rounded decimals.
• 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

-7/2

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.