To use induction you need to prove that the statement is true for initial value(s) of n, and then assume ti is true for n=k to prove that it is true for n=k+1.
In this case, we see that the statement when n=1 is true since `LS=1^2-2^2=-3` and `RS=-1(3)=-3` .
Assume that `1^2-2^2+3^2-4^2+ldots-(2k)^2=-k(2k+1)` is true for some `n=k` .
Now let `n=k+1`.
Then we need to show that
Start with the left side and work to the right side.
`=-k(2k+1)+(2k+1)^2-(2k+2)^2` using the induction assumption
By induction the result has now been proven.