Question

the \triangle\ that pascal studied and published in his treatise was actually more like a truncated corner of tartaglias rectangle. each number in tartaglias rectangle can be calculated in various ways. consider the number n to be located anywhere in the array. by checking several locations in the given array, determine how n is related to the sum of all entries in the shaded cells. click the icon to view the truncated corner of tartaglias rectangle. row 8, column 2 8 row 6, column 3 21 row 4, column 4 20 conjecture the relationship between the value of n and the sum of the values of the shaded cells. choose the correct answer below. a. the sum of the values of the shaded cells is n. b. the sum of the values of the shaded cells is n - 1. c. the sum of the values of the shaded cells is n + 1. d. the sum of the values of the shaded cells is 2n.

Explanation:

Step1: Analyze given data

We have three data - points:

• For Row 8, Column 2, \(N = 8\)
• For Row 6, Column 3, \(N = 21\)
• For Row 4, Column 4, \(N = 20\)

We assume the sum of shaded - cells values is \(S\). Since no information about the sum of shaded - cells is given other than the relationship we need to find, we can test the options with these \(N\) values.

Step2: Test option A

If \(S=N\), for the first case when \(N = 8\), if the sum of shaded - cells \(S\) is equal to \(N\), it holds. For \(N = 21\), if \(S = 21\), it holds. For \(N=20\), if \(S = 20\), it holds.

Step3: Test option B

If \(S=N - 1\), for \(N = 8\), \(S=8 - 1=7
eq8\), so this option is incorrect.

Step4: Test option C

If \(S=N + 1\), for \(N = 8\), \(S=8 + 1=9
eq8\), so this option is incorrect.

Step5: Test option D

If \(S = 2N\), for \(N = 8\), \(S=2\times8 = 16
eq8\), so this option is incorrect.

Answer:

A. The sum of the values of the shaded cells is N.