Question 32609: Finite Left-Endpoint Sum of an Affine Function

Original question and givens

On all R, f(x)=(9)x+(-4). For a=-2≤b=2 and N=9 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)=(9)x+(-4). For a=-2≤b=2 and N=9 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(2, -2)

Left sum ORIGINAL upper minus lower endpoint = 4

Left sum ORIGINAL width over positive cell count := div(4, 9)

Left sum ORIGINAL width over positive cell count = 4/9

Left sum ORIGINAL sample0 index times width := mul(0, 4/9)

Left sum ORIGINAL sample0 index times width = 0

Left sum ORIGINAL sample0 lower plus offset := sum(-2, 0)

Left sum ORIGINAL sample0 lower plus offset = -2

Left sum ORIGINAL sample0 slope times x := mul(9, -2)

Left sum ORIGINAL sample0 slope times x = -18

Left sum ORIGINAL sample0 affine value plus c := sum(-18, -4)

Left sum ORIGINAL sample0 affine value plus c = -22

Left sum ORIGINAL sample0 value times cell width := mul(-22, 4/9)

Left sum ORIGINAL sample0 value times cell width = -88/9

Left sum ORIGINAL accumulated term0 := sum(0, -88/9)

Left sum ORIGINAL accumulated term0 = -88/9

source roles = [0, 9, 9, -4, -2, 2, 4/9]

derived = [-2/1, -22/1, -88/9, -88/9]

Left sum ORIGINAL sample1 index times width := mul(1, 4/9)

Left sum ORIGINAL sample1 index times width = 4/9

Left sum ORIGINAL sample1 lower plus offset := sum(-2, 4/9)

Left sum ORIGINAL sample1 lower plus offset = -14/9

Left sum ORIGINAL sample1 slope times x := mul(9, -14/9)

Left sum ORIGINAL sample1 slope times x = -14

Left sum ORIGINAL sample1 affine value plus c := sum(-14, -4)

Left sum ORIGINAL sample1 affine value plus c = -18

Left sum ORIGINAL sample1 value times cell width := mul(-18, 4/9)

Left sum ORIGINAL sample1 value times cell width = -8

Left sum ORIGINAL accumulated term1 := sum(-88/9, -8)

Left sum ORIGINAL accumulated term1 = -160/9

source roles = [1, 9, 9, -4, -2, 2, 4/9]

derived = [-14/9, -18/1, -8/1, -160/9]

Left sum ORIGINAL sample2 index times width := mul(2, 4/9)

Left sum ORIGINAL sample2 index times width = 8/9

Left sum ORIGINAL sample2 lower plus offset := sum(-2, 8/9)

Left sum ORIGINAL sample2 lower plus offset = -10/9

Left sum ORIGINAL sample2 slope times x := mul(9, -10/9)

Left sum ORIGINAL sample2 slope times x = -10

Left sum ORIGINAL sample2 affine value plus c := sum(-10, -4)

Left sum ORIGINAL sample2 affine value plus c = -14

Left sum ORIGINAL sample2 value times cell width := mul(-14, 4/9)

Left sum ORIGINAL sample2 value times cell width = -56/9

Left sum ORIGINAL accumulated term2 := sum(-160/9, -56/9)

Left sum ORIGINAL accumulated term2 = -24

source roles = [2, 9, 9, -4, -2, 2, 4/9]

derived = [-10/9, -14/1, -56/9, -24/1]

Left sum ORIGINAL sample3 index times width := mul(3, 4/9)

Left sum ORIGINAL sample3 index times width = 4/3

Left sum ORIGINAL sample3 lower plus offset := sum(-2, 4/3)

Left sum ORIGINAL sample3 lower plus offset = -2/3

Left sum ORIGINAL sample3 slope times x := mul(9, -2/3)

Left sum ORIGINAL sample3 slope times x = -6

Left sum ORIGINAL sample3 affine value plus c := sum(-6, -4)

Left sum ORIGINAL sample3 affine value plus c = -10

Left sum ORIGINAL sample3 value times cell width := mul(-10, 4/9)

Left sum ORIGINAL sample3 value times cell width = -40/9

Left sum ORIGINAL accumulated term3 := sum(-24, -40/9)

Left sum ORIGINAL accumulated term3 = -256/9

source roles = [3, 9, 9, -4, -2, 2, 4/9]

derived = [-2/3, -10/1, -40/9, -256/9]

Left sum ORIGINAL sample4 index times width := mul(4, 4/9)

Left sum ORIGINAL sample4 index times width = 16/9

Left sum ORIGINAL sample4 lower plus offset := sum(-2, 16/9)

Left sum ORIGINAL sample4 lower plus offset = -2/9

Left sum ORIGINAL sample4 slope times x := mul(9, -2/9)

Left sum ORIGINAL sample4 slope times x = -2

Left sum ORIGINAL sample4 affine value plus c := sum(-2, -4)

Left sum ORIGINAL sample4 affine value plus c = -6

Left sum ORIGINAL sample4 value times cell width := mul(-6, 4/9)

Left sum ORIGINAL sample4 value times cell width = -8/3

Left sum ORIGINAL accumulated term4 := sum(-256/9, -8/3)

Left sum ORIGINAL accumulated term4 = -280/9

source roles = [4, 9, 9, -4, -2, 2, 4/9]

derived = [-2/9, -6/1, -8/3, -280/9]

Left sum ORIGINAL sample5 index times width := mul(5, 4/9)

Left sum ORIGINAL sample5 index times width = 20/9

Left sum ORIGINAL sample5 lower plus offset := sum(-2, 20/9)

Left sum ORIGINAL sample5 lower plus offset = 2/9

Left sum ORIGINAL sample5 slope times x := mul(9, 2/9)

Left sum ORIGINAL sample5 slope times x = 2

Left sum ORIGINAL sample5 affine value plus c := sum(2, -4)

Left sum ORIGINAL sample5 affine value plus c = -2

Left sum ORIGINAL sample5 value times cell width := mul(-2, 4/9)

Left sum ORIGINAL sample5 value times cell width = -8/9

Left sum ORIGINAL accumulated term5 := sum(-280/9, -8/9)

Left sum ORIGINAL accumulated term5 = -32

source roles = [5, 9, 9, -4, -2, 2, 4/9]

derived = [2/9, -2/1, -8/9, -32/1]

Left sum ORIGINAL sample6 index times width := mul(6, 4/9)

Left sum ORIGINAL sample6 index times width = 8/3

Left sum ORIGINAL sample6 lower plus offset := sum(-2, 8/3)

Left sum ORIGINAL sample6 lower plus offset = 2/3

Left sum ORIGINAL sample6 slope times x := mul(9, 2/3)

Left sum ORIGINAL sample6 slope times x = 6

Left sum ORIGINAL sample6 affine value plus c := sum(6, -4)

Left sum ORIGINAL sample6 affine value plus c = 2

Left sum ORIGINAL sample6 value times cell width := mul(2, 4/9)

Left sum ORIGINAL sample6 value times cell width = 8/9

Left sum ORIGINAL accumulated term6 := sum(-32, 8/9)

Left sum ORIGINAL accumulated term6 = -280/9

source roles = [6, 9, 9, -4, -2, 2, 4/9]

derived = [2/3, 2/1, 8/9, -280/9]

Left sum ORIGINAL sample7 index times width := mul(7, 4/9)

Left sum ORIGINAL sample7 index times width = 28/9

Left sum ORIGINAL sample7 lower plus offset := sum(-2, 28/9)

Left sum ORIGINAL sample7 lower plus offset = 10/9

Left sum ORIGINAL sample7 slope times x := mul(9, 10/9)

Left sum ORIGINAL sample7 slope times x = 10

Left sum ORIGINAL sample7 affine value plus c := sum(10, -4)

Left sum ORIGINAL sample7 affine value plus c = 6

Left sum ORIGINAL sample7 value times cell width := mul(6, 4/9)

Left sum ORIGINAL sample7 value times cell width = 8/3

Left sum ORIGINAL accumulated term7 := sum(-280/9, 8/3)

Left sum ORIGINAL accumulated term7 = -256/9

source roles = [7, 9, 9, -4, -2, 2, 4/9]

derived = [10/9, 6/1, 8/3, -256/9]

Left sum ORIGINAL sample8 index times width := mul(8, 4/9)

Left sum ORIGINAL sample8 index times width = 32/9

Left sum ORIGINAL sample8 lower plus offset := sum(-2, 32/9)

Left sum ORIGINAL sample8 lower plus offset = 14/9

Left sum ORIGINAL sample8 slope times x := mul(9, 14/9)

Left sum ORIGINAL sample8 slope times x = 14

Left sum ORIGINAL sample8 affine value plus c := sum(14, -4)

Left sum ORIGINAL sample8 affine value plus c = 10

Left sum ORIGINAL sample8 value times cell width := mul(10, 4/9)

Left sum ORIGINAL sample8 value times cell width = 40/9

Left sum ORIGINAL accumulated term8 := sum(-256/9, 40/9)

Left sum ORIGINAL accumulated term8 = -24

source roles = [8, 9, 9, -4, -2, 2, 4/9]

derived = [14/9, 10/1, 40/9, -24/1]

source roles = [9, 9, -4, -2, 2]

derived = [4/9, -24/1]

Reverse checks

• S067 · 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.
• S066 · 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.
• S065 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term8 with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 value times cell width with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 affine value plus c with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 slope times x with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 lower plus offset with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 index times width with exact rational arithmetic, not rounded decimals.
• 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

-24/1

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.