If ∑ k = 1 10 k k 4 + k 2 + 1 = m n , where m and n are co-prime, then m + n is equal to

Step-by-step solution

Given, ∑ k = 1 10 k k 4 + k 2 + 1 = m n ⇒ 1 2 ∑ k = 1 10 k 2 + k + 1 - k 2 - k + 1 k 2 + k + 1 k 2 - k + 1 = m n by using the formula  x 4 + x 2 + 1 = x 2 + x + 1 x 2 - x + 1 ⇒ 1 2 ∑ k = 1 10 1 k 2 - k + 1 - 1 k 2 + k + 1 = m n ⇒ 1 2 1 - 1 3 + 1 3 - 1 7 + 1 7 - . . . . . . . . . - 1 111 = m n ⇒ 1 2 1 - 1 111 = m n ⇒ 55 111 = m n So, m + n = 166