Question 32623: Finite Left-Endpoint Sum of an Affine Function

Original question and givens

On all R, f(x)=(23)x+(10). For a=0≤b=3 and N=23 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)=(23)x+(10). For a=0≤b=3 and N=23 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(3, 0)

Left sum ORIGINAL upper minus lower endpoint = 3

Left sum ORIGINAL width over positive cell count := div(3, 23)

Left sum ORIGINAL width over positive cell count = 3/23

Left sum ORIGINAL sample0 index times width := mul(0, 3/23)

Left sum ORIGINAL sample0 index times width = 0

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

Left sum ORIGINAL sample0 lower plus offset = 0

Left sum ORIGINAL sample0 slope times x := mul(23, 0)

Left sum ORIGINAL sample0 slope times x = 0

Left sum ORIGINAL sample0 affine value plus c := sum(0, 10)

Left sum ORIGINAL sample0 affine value plus c = 10

Left sum ORIGINAL sample0 value times cell width := mul(10, 3/23)

Left sum ORIGINAL sample0 value times cell width = 30/23

Left sum ORIGINAL accumulated term0 := sum(0, 30/23)

Left sum ORIGINAL accumulated term0 = 30/23

source roles = [0, 23, 23, 10, 0, 3, 3/23]

derived = [0/1, 10/1, 30/23, 30/23]

Left sum ORIGINAL sample1 index times width := mul(1, 3/23)

Left sum ORIGINAL sample1 index times width = 3/23

Left sum ORIGINAL sample1 lower plus offset := sum(0, 3/23)

Left sum ORIGINAL sample1 lower plus offset = 3/23

Left sum ORIGINAL sample1 slope times x := mul(23, 3/23)

Left sum ORIGINAL sample1 slope times x = 3

Left sum ORIGINAL sample1 affine value plus c := sum(3, 10)

Left sum ORIGINAL sample1 affine value plus c = 13

Left sum ORIGINAL sample1 value times cell width := mul(13, 3/23)

Left sum ORIGINAL sample1 value times cell width = 39/23

Left sum ORIGINAL accumulated term1 := sum(30/23, 39/23)

Left sum ORIGINAL accumulated term1 = 3

source roles = [1, 23, 23, 10, 0, 3, 3/23]

derived = [3/23, 13/1, 39/23, 3/1]

Left sum ORIGINAL sample2 index times width := mul(2, 3/23)

Left sum ORIGINAL sample2 index times width = 6/23

Left sum ORIGINAL sample2 lower plus offset := sum(0, 6/23)

Left sum ORIGINAL sample2 lower plus offset = 6/23

Left sum ORIGINAL sample2 slope times x := mul(23, 6/23)

Left sum ORIGINAL sample2 slope times x = 6

Left sum ORIGINAL sample2 affine value plus c := sum(6, 10)

Left sum ORIGINAL sample2 affine value plus c = 16

Left sum ORIGINAL sample2 value times cell width := mul(16, 3/23)

Left sum ORIGINAL sample2 value times cell width = 48/23

Left sum ORIGINAL accumulated term2 := sum(3, 48/23)

Left sum ORIGINAL accumulated term2 = 117/23

source roles = [2, 23, 23, 10, 0, 3, 3/23]

derived = [6/23, 16/1, 48/23, 117/23]

Left sum ORIGINAL sample3 index times width := mul(3, 3/23)

Left sum ORIGINAL sample3 index times width = 9/23

Left sum ORIGINAL sample3 lower plus offset := sum(0, 9/23)

Left sum ORIGINAL sample3 lower plus offset = 9/23

Left sum ORIGINAL sample3 slope times x := mul(23, 9/23)

Left sum ORIGINAL sample3 slope times x = 9

Left sum ORIGINAL sample3 affine value plus c := sum(9, 10)

Left sum ORIGINAL sample3 affine value plus c = 19

Left sum ORIGINAL sample3 value times cell width := mul(19, 3/23)

Left sum ORIGINAL sample3 value times cell width = 57/23

Left sum ORIGINAL accumulated term3 := sum(117/23, 57/23)

Left sum ORIGINAL accumulated term3 = 174/23

source roles = [3, 23, 23, 10, 0, 3, 3/23]

derived = [9/23, 19/1, 57/23, 174/23]

Left sum ORIGINAL sample4 index times width := mul(4, 3/23)

Left sum ORIGINAL sample4 index times width = 12/23

Left sum ORIGINAL sample4 lower plus offset := sum(0, 12/23)

Left sum ORIGINAL sample4 lower plus offset = 12/23

Left sum ORIGINAL sample4 slope times x := mul(23, 12/23)

Left sum ORIGINAL sample4 slope times x = 12

Left sum ORIGINAL sample4 affine value plus c := sum(12, 10)

Left sum ORIGINAL sample4 affine value plus c = 22

Left sum ORIGINAL sample4 value times cell width := mul(22, 3/23)

Left sum ORIGINAL sample4 value times cell width = 66/23

Left sum ORIGINAL accumulated term4 := sum(174/23, 66/23)

Left sum ORIGINAL accumulated term4 = 240/23

source roles = [4, 23, 23, 10, 0, 3, 3/23]

derived = [12/23, 22/1, 66/23, 240/23]

Left sum ORIGINAL sample5 index times width := mul(5, 3/23)

Left sum ORIGINAL sample5 index times width = 15/23

Left sum ORIGINAL sample5 lower plus offset := sum(0, 15/23)

Left sum ORIGINAL sample5 lower plus offset = 15/23

Left sum ORIGINAL sample5 slope times x := mul(23, 15/23)

Left sum ORIGINAL sample5 slope times x = 15

Left sum ORIGINAL sample5 affine value plus c := sum(15, 10)

Left sum ORIGINAL sample5 affine value plus c = 25

Left sum ORIGINAL sample5 value times cell width := mul(25, 3/23)

Left sum ORIGINAL sample5 value times cell width = 75/23

Left sum ORIGINAL accumulated term5 := sum(240/23, 75/23)

Left sum ORIGINAL accumulated term5 = 315/23

source roles = [5, 23, 23, 10, 0, 3, 3/23]

derived = [15/23, 25/1, 75/23, 315/23]

Left sum ORIGINAL sample6 index times width := mul(6, 3/23)

Left sum ORIGINAL sample6 index times width = 18/23

Left sum ORIGINAL sample6 lower plus offset := sum(0, 18/23)

Left sum ORIGINAL sample6 lower plus offset = 18/23

Left sum ORIGINAL sample6 slope times x := mul(23, 18/23)

Left sum ORIGINAL sample6 slope times x = 18

Left sum ORIGINAL sample6 affine value plus c := sum(18, 10)

Left sum ORIGINAL sample6 affine value plus c = 28

Left sum ORIGINAL sample6 value times cell width := mul(28, 3/23)

Left sum ORIGINAL sample6 value times cell width = 84/23

Left sum ORIGINAL accumulated term6 := sum(315/23, 84/23)

Left sum ORIGINAL accumulated term6 = 399/23

source roles = [6, 23, 23, 10, 0, 3, 3/23]

derived = [18/23, 28/1, 84/23, 399/23]

Left sum ORIGINAL sample7 index times width := mul(7, 3/23)

Left sum ORIGINAL sample7 index times width = 21/23

Left sum ORIGINAL sample7 lower plus offset := sum(0, 21/23)

Left sum ORIGINAL sample7 lower plus offset = 21/23

Left sum ORIGINAL sample7 slope times x := mul(23, 21/23)

Left sum ORIGINAL sample7 slope times x = 21

Left sum ORIGINAL sample7 affine value plus c := sum(21, 10)

Left sum ORIGINAL sample7 affine value plus c = 31

Left sum ORIGINAL sample7 value times cell width := mul(31, 3/23)

Left sum ORIGINAL sample7 value times cell width = 93/23

Left sum ORIGINAL accumulated term7 := sum(399/23, 93/23)

Left sum ORIGINAL accumulated term7 = 492/23

source roles = [7, 23, 23, 10, 0, 3, 3/23]

derived = [21/23, 31/1, 93/23, 492/23]

Left sum ORIGINAL sample8 index times width := mul(8, 3/23)

Left sum ORIGINAL sample8 index times width = 24/23

Left sum ORIGINAL sample8 lower plus offset := sum(0, 24/23)

Left sum ORIGINAL sample8 lower plus offset = 24/23

Left sum ORIGINAL sample8 slope times x := mul(23, 24/23)

Left sum ORIGINAL sample8 slope times x = 24

Left sum ORIGINAL sample8 affine value plus c := sum(24, 10)

Left sum ORIGINAL sample8 affine value plus c = 34

Left sum ORIGINAL sample8 value times cell width := mul(34, 3/23)

Left sum ORIGINAL sample8 value times cell width = 102/23

Left sum ORIGINAL accumulated term8 := sum(492/23, 102/23)

Left sum ORIGINAL accumulated term8 = 594/23

source roles = [8, 23, 23, 10, 0, 3, 3/23]

derived = [24/23, 34/1, 102/23, 594/23]

Left sum ORIGINAL sample9 index times width := mul(9, 3/23)

Left sum ORIGINAL sample9 index times width = 27/23

Left sum ORIGINAL sample9 lower plus offset := sum(0, 27/23)

Left sum ORIGINAL sample9 lower plus offset = 27/23

Left sum ORIGINAL sample9 slope times x := mul(23, 27/23)

Left sum ORIGINAL sample9 slope times x = 27

Left sum ORIGINAL sample9 affine value plus c := sum(27, 10)

Left sum ORIGINAL sample9 affine value plus c = 37

Left sum ORIGINAL sample9 value times cell width := mul(37, 3/23)

Left sum ORIGINAL sample9 value times cell width = 111/23

Left sum ORIGINAL accumulated term9 := sum(594/23, 111/23)

Left sum ORIGINAL accumulated term9 = 705/23

source roles = [9, 23, 23, 10, 0, 3, 3/23]

derived = [27/23, 37/1, 111/23, 705/23]

Left sum ORIGINAL sample10 index times width := mul(10, 3/23)

Left sum ORIGINAL sample10 index times width = 30/23

Left sum ORIGINAL sample10 lower plus offset := sum(0, 30/23)

Left sum ORIGINAL sample10 lower plus offset = 30/23

Left sum ORIGINAL sample10 slope times x := mul(23, 30/23)

Left sum ORIGINAL sample10 slope times x = 30

Left sum ORIGINAL sample10 affine value plus c := sum(30, 10)

Left sum ORIGINAL sample10 affine value plus c = 40

Left sum ORIGINAL sample10 value times cell width := mul(40, 3/23)

Left sum ORIGINAL sample10 value times cell width = 120/23

Left sum ORIGINAL accumulated term10 := sum(705/23, 120/23)

Left sum ORIGINAL accumulated term10 = 825/23

source roles = [10, 23, 23, 10, 0, 3, 3/23]

derived = [30/23, 40/1, 120/23, 825/23]

Left sum ORIGINAL sample11 index times width := mul(11, 3/23)

Left sum ORIGINAL sample11 index times width = 33/23

Left sum ORIGINAL sample11 lower plus offset := sum(0, 33/23)

Left sum ORIGINAL sample11 lower plus offset = 33/23

Left sum ORIGINAL sample11 slope times x := mul(23, 33/23)

Left sum ORIGINAL sample11 slope times x = 33

Left sum ORIGINAL sample11 affine value plus c := sum(33, 10)

Left sum ORIGINAL sample11 affine value plus c = 43

Left sum ORIGINAL sample11 value times cell width := mul(43, 3/23)

Left sum ORIGINAL sample11 value times cell width = 129/23

Left sum ORIGINAL accumulated term11 := sum(825/23, 129/23)

Left sum ORIGINAL accumulated term11 = 954/23

source roles = [11, 23, 23, 10, 0, 3, 3/23]

derived = [33/23, 43/1, 129/23, 954/23]

Left sum ORIGINAL sample12 index times width := mul(12, 3/23)

Left sum ORIGINAL sample12 index times width = 36/23

Left sum ORIGINAL sample12 lower plus offset := sum(0, 36/23)

Left sum ORIGINAL sample12 lower plus offset = 36/23

Left sum ORIGINAL sample12 slope times x := mul(23, 36/23)

Left sum ORIGINAL sample12 slope times x = 36

Left sum ORIGINAL sample12 affine value plus c := sum(36, 10)

Left sum ORIGINAL sample12 affine value plus c = 46

Left sum ORIGINAL sample12 value times cell width := mul(46, 3/23)

Left sum ORIGINAL sample12 value times cell width = 6

Left sum ORIGINAL accumulated term12 := sum(954/23, 6)

Left sum ORIGINAL accumulated term12 = 1092/23

source roles = [12, 23, 23, 10, 0, 3, 3/23]

derived = [36/23, 46/1, 6/1, 1092/23]

Left sum ORIGINAL sample13 index times width := mul(13, 3/23)

Left sum ORIGINAL sample13 index times width = 39/23

Left sum ORIGINAL sample13 lower plus offset := sum(0, 39/23)

Left sum ORIGINAL sample13 lower plus offset = 39/23

Left sum ORIGINAL sample13 slope times x := mul(23, 39/23)

Left sum ORIGINAL sample13 slope times x = 39

Left sum ORIGINAL sample13 affine value plus c := sum(39, 10)

Left sum ORIGINAL sample13 affine value plus c = 49

Left sum ORIGINAL sample13 value times cell width := mul(49, 3/23)

Left sum ORIGINAL sample13 value times cell width = 147/23

Left sum ORIGINAL accumulated term13 := sum(1092/23, 147/23)

Left sum ORIGINAL accumulated term13 = 1239/23

source roles = [13, 23, 23, 10, 0, 3, 3/23]

derived = [39/23, 49/1, 147/23, 1239/23]

Left sum ORIGINAL sample14 index times width := mul(14, 3/23)

Left sum ORIGINAL sample14 index times width = 42/23

Left sum ORIGINAL sample14 lower plus offset := sum(0, 42/23)

Left sum ORIGINAL sample14 lower plus offset = 42/23

Left sum ORIGINAL sample14 slope times x := mul(23, 42/23)

Left sum ORIGINAL sample14 slope times x = 42

Left sum ORIGINAL sample14 affine value plus c := sum(42, 10)

Left sum ORIGINAL sample14 affine value plus c = 52

Left sum ORIGINAL sample14 value times cell width := mul(52, 3/23)

Left sum ORIGINAL sample14 value times cell width = 156/23

Left sum ORIGINAL accumulated term14 := sum(1239/23, 156/23)

Left sum ORIGINAL accumulated term14 = 1395/23

source roles = [14, 23, 23, 10, 0, 3, 3/23]

derived = [42/23, 52/1, 156/23, 1395/23]

Left sum ORIGINAL sample15 index times width := mul(15, 3/23)

Left sum ORIGINAL sample15 index times width = 45/23

Left sum ORIGINAL sample15 lower plus offset := sum(0, 45/23)

Left sum ORIGINAL sample15 lower plus offset = 45/23

Left sum ORIGINAL sample15 slope times x := mul(23, 45/23)

Left sum ORIGINAL sample15 slope times x = 45

Left sum ORIGINAL sample15 affine value plus c := sum(45, 10)

Left sum ORIGINAL sample15 affine value plus c = 55

Left sum ORIGINAL sample15 value times cell width := mul(55, 3/23)

Left sum ORIGINAL sample15 value times cell width = 165/23

Left sum ORIGINAL accumulated term15 := sum(1395/23, 165/23)

Left sum ORIGINAL accumulated term15 = 1560/23

source roles = [15, 23, 23, 10, 0, 3, 3/23]

derived = [45/23, 55/1, 165/23, 1560/23]

Left sum ORIGINAL sample16 index times width := mul(16, 3/23)

Left sum ORIGINAL sample16 index times width = 48/23

Left sum ORIGINAL sample16 lower plus offset := sum(0, 48/23)

Left sum ORIGINAL sample16 lower plus offset = 48/23

Left sum ORIGINAL sample16 slope times x := mul(23, 48/23)

Left sum ORIGINAL sample16 slope times x = 48

Left sum ORIGINAL sample16 affine value plus c := sum(48, 10)

Left sum ORIGINAL sample16 affine value plus c = 58

Left sum ORIGINAL sample16 value times cell width := mul(58, 3/23)

Left sum ORIGINAL sample16 value times cell width = 174/23

Left sum ORIGINAL accumulated term16 := sum(1560/23, 174/23)

Left sum ORIGINAL accumulated term16 = 1734/23

source roles = [16, 23, 23, 10, 0, 3, 3/23]

derived = [48/23, 58/1, 174/23, 1734/23]

Left sum ORIGINAL sample17 index times width := mul(17, 3/23)

Left sum ORIGINAL sample17 index times width = 51/23

Left sum ORIGINAL sample17 lower plus offset := sum(0, 51/23)

Left sum ORIGINAL sample17 lower plus offset = 51/23

Left sum ORIGINAL sample17 slope times x := mul(23, 51/23)

Left sum ORIGINAL sample17 slope times x = 51

Left sum ORIGINAL sample17 affine value plus c := sum(51, 10)

Left sum ORIGINAL sample17 affine value plus c = 61

Left sum ORIGINAL sample17 value times cell width := mul(61, 3/23)

Left sum ORIGINAL sample17 value times cell width = 183/23

Left sum ORIGINAL accumulated term17 := sum(1734/23, 183/23)

Left sum ORIGINAL accumulated term17 = 1917/23

source roles = [17, 23, 23, 10, 0, 3, 3/23]

derived = [51/23, 61/1, 183/23, 1917/23]

Left sum ORIGINAL sample18 index times width := mul(18, 3/23)

Left sum ORIGINAL sample18 index times width = 54/23

Left sum ORIGINAL sample18 lower plus offset := sum(0, 54/23)

Left sum ORIGINAL sample18 lower plus offset = 54/23

Left sum ORIGINAL sample18 slope times x := mul(23, 54/23)

Left sum ORIGINAL sample18 slope times x = 54

Left sum ORIGINAL sample18 affine value plus c := sum(54, 10)

Left sum ORIGINAL sample18 affine value plus c = 64

Left sum ORIGINAL sample18 value times cell width := mul(64, 3/23)

Left sum ORIGINAL sample18 value times cell width = 192/23

Left sum ORIGINAL accumulated term18 := sum(1917/23, 192/23)

Left sum ORIGINAL accumulated term18 = 2109/23

source roles = [18, 23, 23, 10, 0, 3, 3/23]

derived = [54/23, 64/1, 192/23, 2109/23]

Left sum ORIGINAL sample19 index times width := mul(19, 3/23)

Left sum ORIGINAL sample19 index times width = 57/23

Left sum ORIGINAL sample19 lower plus offset := sum(0, 57/23)

Left sum ORIGINAL sample19 lower plus offset = 57/23

Left sum ORIGINAL sample19 slope times x := mul(23, 57/23)

Left sum ORIGINAL sample19 slope times x = 57

Left sum ORIGINAL sample19 affine value plus c := sum(57, 10)

Left sum ORIGINAL sample19 affine value plus c = 67

Left sum ORIGINAL sample19 value times cell width := mul(67, 3/23)

Left sum ORIGINAL sample19 value times cell width = 201/23

Left sum ORIGINAL accumulated term19 := sum(2109/23, 201/23)

Left sum ORIGINAL accumulated term19 = 2310/23

source roles = [19, 23, 23, 10, 0, 3, 3/23]

derived = [57/23, 67/1, 201/23, 2310/23]

Left sum ORIGINAL sample20 index times width := mul(20, 3/23)

Left sum ORIGINAL sample20 index times width = 60/23

Left sum ORIGINAL sample20 lower plus offset := sum(0, 60/23)

Left sum ORIGINAL sample20 lower plus offset = 60/23

Left sum ORIGINAL sample20 slope times x := mul(23, 60/23)

Left sum ORIGINAL sample20 slope times x = 60

Left sum ORIGINAL sample20 affine value plus c := sum(60, 10)

Left sum ORIGINAL sample20 affine value plus c = 70

Left sum ORIGINAL sample20 value times cell width := mul(70, 3/23)

Left sum ORIGINAL sample20 value times cell width = 210/23

Left sum ORIGINAL accumulated term20 := sum(2310/23, 210/23)

Left sum ORIGINAL accumulated term20 = 2520/23

source roles = [20, 23, 23, 10, 0, 3, 3/23]

derived = [60/23, 70/1, 210/23, 2520/23]

Left sum ORIGINAL sample21 index times width := mul(21, 3/23)

Left sum ORIGINAL sample21 index times width = 63/23

Left sum ORIGINAL sample21 lower plus offset := sum(0, 63/23)

Left sum ORIGINAL sample21 lower plus offset = 63/23

Left sum ORIGINAL sample21 slope times x := mul(23, 63/23)

Left sum ORIGINAL sample21 slope times x = 63

Left sum ORIGINAL sample21 affine value plus c := sum(63, 10)

Left sum ORIGINAL sample21 affine value plus c = 73

Left sum ORIGINAL sample21 value times cell width := mul(73, 3/23)

Left sum ORIGINAL sample21 value times cell width = 219/23

Left sum ORIGINAL accumulated term21 := sum(2520/23, 219/23)

Left sum ORIGINAL accumulated term21 = 2739/23

source roles = [21, 23, 23, 10, 0, 3, 3/23]

derived = [63/23, 73/1, 219/23, 2739/23]

Left sum ORIGINAL sample22 index times width := mul(22, 3/23)

Left sum ORIGINAL sample22 index times width = 66/23

Left sum ORIGINAL sample22 lower plus offset := sum(0, 66/23)

Left sum ORIGINAL sample22 lower plus offset = 66/23

Left sum ORIGINAL sample22 slope times x := mul(23, 66/23)

Left sum ORIGINAL sample22 slope times x = 66

Left sum ORIGINAL sample22 affine value plus c := sum(66, 10)

Left sum ORIGINAL sample22 affine value plus c = 76

Left sum ORIGINAL sample22 value times cell width := mul(76, 3/23)

Left sum ORIGINAL sample22 value times cell width = 228/23

Left sum ORIGINAL accumulated term22 := sum(2739/23, 228/23)

Left sum ORIGINAL accumulated term22 = 129

source roles = [22, 23, 23, 10, 0, 3, 3/23]

derived = [66/23, 76/1, 228/23, 129/1]

source roles = [23, 23, 10, 0, 3]

derived = [3/23, 129/1]

Reverse checks

• S165 · 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.
• S164 · 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.
• S163 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term22 with exact rational arithmetic, not rounded decimals.
• S162 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 value times cell width with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 affine value plus c with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 slope times x with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 lower plus offset with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 index times width with exact rational arithmetic, not rounded decimals.
• S157 · 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.
• S156 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term21 with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 value times cell width with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 affine value plus c with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 slope times x with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 lower plus offset with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 index times width with exact rational arithmetic, not rounded decimals.
• S150 · 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.
• S149 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term20 with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 value times cell width with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 affine value plus c with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 slope times x with exact rational arithmetic, not rounded decimals.
• S145 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 lower plus offset with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 index times width with exact rational arithmetic, not rounded decimals.
• S143 · 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.
• S142 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term19 with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 value times cell width with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 affine value plus c with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 slope times x with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 lower plus offset with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 index times width with exact rational arithmetic, not rounded decimals.
• S136 · 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.
• S135 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term18 with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 value times cell width with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 affine value plus c with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 slope times x with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 lower plus offset with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 index times width with exact rational arithmetic, not rounded decimals.
• S129 · 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.
• S128 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term17 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 value times cell width with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 affine value plus c with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 slope times x with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 lower plus offset with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 index times width with exact rational arithmetic, not rounded decimals.
• S122 · 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.
• S121 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term16 with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 value times cell width with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 affine value plus c with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 slope times x with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 lower plus offset with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 index times width with exact rational arithmetic, not rounded decimals.
• S115 · 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.
• S114 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term15 with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 value times cell width with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 affine value plus c with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 slope times x with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 lower plus offset with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 index times width with exact rational arithmetic, not rounded decimals.
• S108 · 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.
• S107 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term14 with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 value times cell width with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 affine value plus c with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 slope times x with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 lower plus offset with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 index times width with exact rational arithmetic, not rounded decimals.
• S101 · 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.
• S100 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term13 with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 value times cell width with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 affine value plus c with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 slope times x with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 lower plus offset with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 index times width with exact rational arithmetic, not rounded decimals.
• S094 · 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.
• S093 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term12 with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 value times cell width with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 affine value plus c with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 slope times x with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 lower plus offset with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 index times width with exact rational arithmetic, not rounded decimals.
• S087 · 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.
• S086 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term11 with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 value times cell width with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 affine value plus c with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 slope times x with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 lower plus offset with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 index times width with exact rational arithmetic, not rounded decimals.
• S080 · 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.
• S079 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term10 with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 value times cell width with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 affine value plus c with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 slope times x with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 lower plus offset with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 index times width with exact rational arithmetic, not rounded decimals.
• S073 · 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.
• S072 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term9 with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 value times cell width with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 affine value plus c with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 slope times x with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 lower plus offset with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 index times width with exact rational arithmetic, not rounded decimals.
• 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

129/1

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.