<s>[INST] Proving that the sum of the first $2n$ terms of the series $1^2 - 3^2 + 5^2 - \cdots$ is $-8n^2$ by induction 
Use mathematical induction to prove the following for the first $2n$ terms of the series $$1^2 - 3^2 + 5^2 - 7^2 + \cdots = -8n^2.$$

As we have odd numbers that are squared we could use $n = 2k-1$.
But the $2$ sides do not equate for $n=1$ or $2k-1$ (if you set $k=1$).
Also need to find the sum to $2n+1$ terms.
 [/INST] 
As we have odd numbers that are squared we could use n = 2k-1

You are summing the sequence:  $\{1^2-3^2, 5^2-7^2, 9^2-11^2, \ldots (4n-3)^2-(4n-1)^2, \ldots\}$

But the 2 sides do not equate for n=1 or 2k-1 (if you set k=1)

See above. The first term is $-8$, and so...

Also need to find the sum to (2n+1) terms.

You need to show that if the sum of the first $n$ terms is $-8 n^2$, then if you add $-16(n+1)+8$ you obtain $-8(n+1)^2$.
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