If the k^ th term of a sequence is the ratio of the sum of the squares of the first k natural numbers to the sum of the first k natural numbers, and the sum of the first n terms of this sequence is 85 , then the value of n is :

Step-by-step solution

Let the k^ th term of the sequence be T_k . According to the given condition: T_k = _ i=1 ^ k i^2 _ i=1 ^ k i Using the standard formulas for the sum of squares and the sum of the first k natural numbers: T_k = k(k+1)(2k+1) 6 k(k+1) 2 T_k = 2k+1 3 The sum of the first n terms is S_n = _ k=1 ^ n T_k S_n = _ k=1 ^ n 2k+1 3 = 1 3 ( 2 _ k=1 ^ n k + _ k=1 ^ n 1 ) S_n = 1 3 ( 2 n(n+1) 2 + n ) = n(n+1) + n 3 S_n = n^2 + 2n 3 = n(n+2) 3 We are given that S_n = 85 . Equating the two expressions: n(n+2) 3 = 85 n(n+2) = 255 n