If the sum of the first n terms of an arithmetic progression, whose first term is the sum of the first n positive integers and whose common difference is n , is 8 n 2 - 11 n - 20 , then the sum of all the possible values of n is

Step-by-step solution

a 1 = n n + 1 2 , d = n ⇒ S n = n 2 n n + 1 + n - 1 n = n 2 n 2 + n + n 2 - n = n 3 But, given that S n = 8 n 2 - 11 n - 20 . ⇒ 8 n 2 - 11 n - 20 = n 3 ⇒ n 3 - 8 n 2 + 11 n + 20 = 0 ⇒ n + 1 n - 4 n - 5 = 0 ⇒ n = 4 or 5 n ≠ - 1 ⇒ sum of all possible values = 9