I have a following problem
When we want to write $a^2 + b^2$ in terms of $(a \pm b)^2$ we can do it like that $a^2 +b^2 = \frac{(a+b)^2}{2} + \frac{(a-b)^2}{2}.$
Can we do anything similar for $a_1^2 + a_2^2 + \ldots + a_n^2$ ? I can add the assumption that all $a_i$ are positive numbers. I mean to express this as combination of their sums and differences. I know that this question is a little bit naive but I'm curious whether it has an easy answer.