Let α = 1 2 + 4 2 + 8 2 + 13 2 + 19 2 + 26 2 + … … . upto 10 terms and β = ∑ n = 1 10 n 4 . If 4 α - β = 55 k + 40 , then k is equal to _______.

Step-by-step solution

Given, α = 1 2 + 4 2 + 8 2 … Now, let t n = a n 2 + b n + c 1 = a + b + c . . . . i 4 = 4 a + 2 b + c . . . . i i 8 = 9 a + 3 b + c . . . . i i i Now, solving equation i , i i & i i i we get, a = 1 2 , b = 3 2 , c = - 1 Hence, t n = n 2 2 + 3 n 2 - 1 2 α = ∑ n = 1 10 n 2 2 + 3 n 2 - 1 2 ⇒ 4 α = ∑ n = 1 10 n 2 + 3 n - 2 2 and β = ∑ n = 1 10 n 4 ⇒ 4 α - β = ∑ n = 1 10 6 n 3 + 5 n 2 - 12 n + 4 ⇒ 4 α - β = 6 ∑ n = 1 10 n 3 + 5 ∑ n = 1 10 n 2 - 12 ∑ n = 1 10 n + 4 ∑ n = 1 10 1 ⇒ 4 α - β = 6 10 10 + 1 2 2 + 5 10 · 11 · 2