How to substitute sum variable?

Question

I am not entirely sure how to use variable substitution for a sum. Take the following example: I would like to compute $$\sum_{i=1}^N(2i-1)^2$$ One straightforward way is to split the sum, i.e. $$4\sum i^2-4\sum i+N=\frac{1}{3}N(4N^2-1)$$ Now I am trying to use variable substitution. Let $j=2i-1\Rightarrow i=\frac{j+1}{2}$. The new limits become: $$\frac{j+1}{2}=1\Rightarrow j=1\\\frac{j+1}{2}=N\Rightarrow j=2N-1$$. The sum becomes $$\sum_{j=1}^{2N-1}j^2=\frac{1}{3}n(2n-1)(4n-1)$$ which is wrong :(

Answer 1

Continuing lab's answer: You could write, say $$ \sum_{j=1,\; j\;\mathrm{odd}}^{2N-1} j^2 $$ but of course the formulas you know for the case without the "odd" clause would not apply to it.

Answer 2

Observe that $j$ does not vary from $1$ to $2N-1$ with increment $=1$ but with increment $=2$

so, we can not apply Prove that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?