Help with this summation?

Question

Is there a simplified expression for

{Sum From j = 0 to j = N-1} of (y - x)^2

where y is x with an index of j+1 and x is meant to have an index of j. I hope that's clear enough!!

Answer 1

Without knowing the relationship between successive elements, I think the best you can do is this:
[k=0, n-1]∑(x_(k+1) - x_k)²
[k=0, n-1]∑(x_(k+1)² - 2x_k*x_(k+1) + x_k²)
[k=0, n-1]∑x_(k+1)² - 2[k=0, n-1]∑(x_k*x_(k+1)) + [k=0, n-1]∑x_k²
[k=1, n]∑x_k² - 2[k=0, n-1]∑(x_k*x_(k+1)) + [k=0, n-1]∑x_k²

x_0 + x_n + 2[k=1, n-1]∑x_k² - 2[k=0, n-1]∑(x_k*x_(k+1))

Answer 2

I read this as
N-1
∑(j + 1 - j)^2 = N - 1
j=1