Different A.P.'s are constructed with the first term 100 , the last term 199 , And integral common differences. The sum of the common differences of all such, A.P's having at least 3 terms and at most 33 terms is.

Step-by-step solution

Given 1 st term of the A.P = 100 = a And the last term = 199 = L Let the three term be a , a + d , a + 2 d a n = L = a + n - 1 d d i = L - a n - 1 n → number of terms Now checking for what value of n we have integral common difference, For, n = 3 ,   d 1 = 199 - 100 2 = 99 2 ∉ I For, n = 4 ,   d 2 = 99 3 = 33 ∈ I . . . For, n = 10 ,   d 3 = 99 9 = 11 ∈ I For, n = 12 ,   d 4 = 99 11 = 9 ∈ I So, sum of all common difference that are integer will be ∑ d i = 33 +