Let S_n = _ k=1 ^ 4n (-1)^ k(k+1) 2 k^2 . If S_n = 1056 , then the value of n is _____.

Step-by-step solution

The given sum is S_n = _ k=1 ^ 4n (-1)^ k(k+1) 2 k^2 . Let us analyze the sign pattern of the terms by evaluating (-1)^ k(k+1) 2 for the first few values of k : For k=1 , (-1)^1 = -1 For k=2 , (-1)^3 = -1 For k=3 , (-1)^6 = 1 For k=4 , (-1)¹⁰ = 1 The pattern of signs is -, -, +, + and it repeats every 4 terms. We can group the sum into n blocks of 4 terms. The m -th block consists of the terms for k = 4m-3, 4m-2, 4m-1, 4m . The sum of the terms in the m -th block is: -(4m-3)^2 - (4m-2)^2 + (4m-1)^2 + (4m)^2 Rearran