Finding the sum of the series $\sum\limits_{j>i=1}^{2n}(-1)^{i+j}$

Question

I have to find the sum

$\sum\limits_{j>i=1}^{2n}(-1)^{i+j}$.

Please anyone help me how to solve this problem. Thanks in advance.

Answer 1

One elegant way how to see the result may be this. The sum over all $i,j$ such that $1\leq i,j \leq 2n$ is obviously $0$. Split it to three parts $<$, $=$ and $>$ and you get the equation $x + 2n + x = 0$

Answer 2

As the terms cancel each other in pairs,

$$\sum_{i=1}^{2n}\sum_{j=i+1}^{2n}(-1)^{i+j}=\sum_{i=1}^{2n}\sum_{j=i+1}^{2n}(-1)^{i+j}=-\sum_{i=1}^{2n}(i\bmod2)=-n.$$