For a homework problem, I need to prove $f_0f_1+f_1f_2+...+f_{2n-1}f_{2n}=f_{2n}^2$ for $n\geq1$ with induction. So far, using my basis step, I have $$\sum_{i=1}^{k+1} f_{2(k+1)-1}f_{2(k+1)}=$$ $$\left(\sum_{i=1}^{k}f_{2k-1}f_{2k}\right)+f_{2k+1}f_{2k+2}=$$ $$f_{2k}^2+f_{2k+1}f_{2k+2}$$
However, I'm stuck here. I've tried breaking down the terms into their recursive terms and FOIL-ing/factoring, etc, and I can't seem to get to the end goal of $f_{2k+2}^2$. I might be missing something obvious... Any help in the right direction would be greatly appreciated.