Question 32608: Finite Left-Endpoint Sum of an Affine Function

Original question and givens

On all R, f(x)=(-8)x+(-5). For a=1≤b=4 and N=8 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)=(-8)x+(-5). For a=1≤b=4 and N=8 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(4, 1)

Left sum ORIGINAL upper minus lower endpoint = 3

Left sum ORIGINAL width over positive cell count := div(3, 8)

Left sum ORIGINAL width over positive cell count = 3/8

Left sum ORIGINAL sample0 index times width := mul(0, 3/8)

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(-8, 1)

Left sum ORIGINAL sample0 slope times x = -8

Left sum ORIGINAL sample0 affine value plus c := sum(-8, -5)

Left sum ORIGINAL sample0 affine value plus c = -13

Left sum ORIGINAL sample0 value times cell width := mul(-13, 3/8)

Left sum ORIGINAL sample0 value times cell width = -39/8

Left sum ORIGINAL accumulated term0 := sum(0, -39/8)

Left sum ORIGINAL accumulated term0 = -39/8

source roles = [0, 8, -8, -5, 1, 4, 3/8]

derived = [1/1, -13/1, -39/8, -39/8]

Left sum ORIGINAL sample1 index times width := mul(1, 3/8)

Left sum ORIGINAL sample1 index times width = 3/8

Left sum ORIGINAL sample1 lower plus offset := sum(1, 3/8)

Left sum ORIGINAL sample1 lower plus offset = 11/8

Left sum ORIGINAL sample1 slope times x := mul(-8, 11/8)

Left sum ORIGINAL sample1 slope times x = -11

Left sum ORIGINAL sample1 affine value plus c := sum(-11, -5)

Left sum ORIGINAL sample1 affine value plus c = -16

Left sum ORIGINAL sample1 value times cell width := mul(-16, 3/8)

Left sum ORIGINAL sample1 value times cell width = -6

Left sum ORIGINAL accumulated term1 := sum(-39/8, -6)

Left sum ORIGINAL accumulated term1 = -87/8

source roles = [1, 8, -8, -5, 1, 4, 3/8]

derived = [11/8, -16/1, -6/1, -87/8]

Left sum ORIGINAL sample2 index times width := mul(2, 3/8)

Left sum ORIGINAL sample2 index times width = 3/4

Left sum ORIGINAL sample2 lower plus offset := sum(1, 3/4)

Left sum ORIGINAL sample2 lower plus offset = 7/4

Left sum ORIGINAL sample2 slope times x := mul(-8, 7/4)

Left sum ORIGINAL sample2 slope times x = -14

Left sum ORIGINAL sample2 affine value plus c := sum(-14, -5)

Left sum ORIGINAL sample2 affine value plus c = -19

Left sum ORIGINAL sample2 value times cell width := mul(-19, 3/8)

Left sum ORIGINAL sample2 value times cell width = -57/8

Left sum ORIGINAL accumulated term2 := sum(-87/8, -57/8)

Left sum ORIGINAL accumulated term2 = -18

source roles = [2, 8, -8, -5, 1, 4, 3/8]

derived = [7/4, -19/1, -57/8, -18/1]

Left sum ORIGINAL sample3 index times width := mul(3, 3/8)

Left sum ORIGINAL sample3 index times width = 9/8

Left sum ORIGINAL sample3 lower plus offset := sum(1, 9/8)

Left sum ORIGINAL sample3 lower plus offset = 17/8

Left sum ORIGINAL sample3 slope times x := mul(-8, 17/8)

Left sum ORIGINAL sample3 slope times x = -17

Left sum ORIGINAL sample3 affine value plus c := sum(-17, -5)

Left sum ORIGINAL sample3 affine value plus c = -22

Left sum ORIGINAL sample3 value times cell width := mul(-22, 3/8)

Left sum ORIGINAL sample3 value times cell width = -33/4

Left sum ORIGINAL accumulated term3 := sum(-18, -33/4)

Left sum ORIGINAL accumulated term3 = -105/4

source roles = [3, 8, -8, -5, 1, 4, 3/8]

derived = [17/8, -22/1, -33/4, -105/4]

Left sum ORIGINAL sample4 index times width := mul(4, 3/8)

Left sum ORIGINAL sample4 index times width = 3/2

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

Left sum ORIGINAL sample4 lower plus offset = 5/2

Left sum ORIGINAL sample4 slope times x := mul(-8, 5/2)

Left sum ORIGINAL sample4 slope times x = -20

Left sum ORIGINAL sample4 affine value plus c := sum(-20, -5)

Left sum ORIGINAL sample4 affine value plus c = -25

Left sum ORIGINAL sample4 value times cell width := mul(-25, 3/8)

Left sum ORIGINAL sample4 value times cell width = -75/8

Left sum ORIGINAL accumulated term4 := sum(-105/4, -75/8)

Left sum ORIGINAL accumulated term4 = -285/8

source roles = [4, 8, -8, -5, 1, 4, 3/8]

derived = [5/2, -25/1, -75/8, -285/8]

Left sum ORIGINAL sample5 index times width := mul(5, 3/8)

Left sum ORIGINAL sample5 index times width = 15/8

Left sum ORIGINAL sample5 lower plus offset := sum(1, 15/8)

Left sum ORIGINAL sample5 lower plus offset = 23/8

Left sum ORIGINAL sample5 slope times x := mul(-8, 23/8)

Left sum ORIGINAL sample5 slope times x = -23

Left sum ORIGINAL sample5 affine value plus c := sum(-23, -5)

Left sum ORIGINAL sample5 affine value plus c = -28

Left sum ORIGINAL sample5 value times cell width := mul(-28, 3/8)

Left sum ORIGINAL sample5 value times cell width = -21/2

Left sum ORIGINAL accumulated term5 := sum(-285/8, -21/2)

Left sum ORIGINAL accumulated term5 = -369/8

source roles = [5, 8, -8, -5, 1, 4, 3/8]

derived = [23/8, -28/1, -21/2, -369/8]

Left sum ORIGINAL sample6 index times width := mul(6, 3/8)

Left sum ORIGINAL sample6 index times width = 9/4

Left sum ORIGINAL sample6 lower plus offset := sum(1, 9/4)

Left sum ORIGINAL sample6 lower plus offset = 13/4

Left sum ORIGINAL sample6 slope times x := mul(-8, 13/4)

Left sum ORIGINAL sample6 slope times x = -26

Left sum ORIGINAL sample6 affine value plus c := sum(-26, -5)

Left sum ORIGINAL sample6 affine value plus c = -31

Left sum ORIGINAL sample6 value times cell width := mul(-31, 3/8)

Left sum ORIGINAL sample6 value times cell width = -93/8

Left sum ORIGINAL accumulated term6 := sum(-369/8, -93/8)

Left sum ORIGINAL accumulated term6 = -231/4

source roles = [6, 8, -8, -5, 1, 4, 3/8]

derived = [13/4, -31/1, -93/8, -231/4]

Left sum ORIGINAL sample7 index times width := mul(7, 3/8)

Left sum ORIGINAL sample7 index times width = 21/8

Left sum ORIGINAL sample7 lower plus offset := sum(1, 21/8)

Left sum ORIGINAL sample7 lower plus offset = 29/8

Left sum ORIGINAL sample7 slope times x := mul(-8, 29/8)

Left sum ORIGINAL sample7 slope times x = -29

Left sum ORIGINAL sample7 affine value plus c := sum(-29, -5)

Left sum ORIGINAL sample7 affine value plus c = -34

Left sum ORIGINAL sample7 value times cell width := mul(-34, 3/8)

Left sum ORIGINAL sample7 value times cell width = -51/4

Left sum ORIGINAL accumulated term7 := sum(-231/4, -51/4)

Left sum ORIGINAL accumulated term7 = -141/2

source roles = [7, 8, -8, -5, 1, 4, 3/8]

derived = [29/8, -34/1, -51/4, -141/2]

source roles = [8, -8, -5, 1, 4]

derived = [3/8, -141/2]

Reverse checks

• S060 · 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.
• S059 · 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.
• S058 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term7 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 value times cell width with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 affine value plus c with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 slope times x with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 lower plus offset with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 index times width with exact rational arithmetic, not rounded decimals.
• S052 · 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.
• S051 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term6 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 value times cell width with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 affine value plus c with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 slope times x with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 lower plus offset with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 index times width with exact rational arithmetic, not rounded decimals.
• 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

-141/2

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.