<s>[INST] Proving Fibonacci sum with induction Let the Fibonacci sequence be defined as $f_1 = f_2 = 1$ and $f_n = f_{n-1} + f_{n-2}$
Prove that $f_1f_2+f_2f_3+f_3f_4+...+f_{2n-1}f_{2n}+f_{2n}f_{2n+1} = f^2_{2n+1} - 1$
I was able to verify that the claim holds for $n=1$, since $(1)(1)+(1)(2) = 2^2 - 1$
Now, assuming that the claim is true for every $n=k$, it should be sufficient to prove by checking if $n=k+1$ also is true.
$f_1f_2+f_2f_3+...+f_{2k-1}f_{2k}+f_{2k}f_{2k+1}+f_{2(k+1)}f_{2(k+1)+1}$
$ f^2_{2k+1} - 1 +f_{2(k+1)}f_{2(k+1)+1}$
I'm not sure how to proceed from here.
 [/INST] Hint.
If the statement holds for $n$, then
$f_1f_2 + \dots + f_{2n}f_{2n+1}+ f_{2n+1}f_{2n+2}+f_{2n+2}f_{2n+3}= (\boxed{f_{2n+1}^2}-1) +\boxed{f_{2n+1}f_{2n+2}}+f_{2n+2}f_{2n+3}= f_{2n+1}\underbrace{(f_{2n+1} + f_{2n+2})}_{=\text{what is it?}} + f_{2n+2}f_{2n+3} -1 = \dots \text{(similar step)} = f_{2n+3}^2-1.$
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