Let's start from what we know.
Note that:
(sign of last term will be + when n is even and - when n is odd).
Sum is finite so we can split it into two sums, first
with only positive trems (squares of even numbers) and second 
with negative (squares of odd numbers). So:
And now the proof.
1) n is even.
In this case, both and 
have 
terms. For example if n=8 then:
Generally, there will be:
Now, calculate our sum:
So in this case we prove, that:
2) n is odd.
Here, has more terms than . For example if n=7 then:
So there is
terms in , 
terms in and:
Now, we can calculate our sum:
We consider all possible n so we prove that:
Note that:
(sign of last term will be + when n is even and - when n is odd).
Sum is finite so we can split it into two sums, first
And now the proof.
1) n is even.
In this case, both
Generally, there will be:
Now, calculate our sum:
So in this case we prove, that:
2) n is odd.
Here,
So there is
Now, we can calculate our sum:
We consider all possible n so we prove that: