I am having troubles with a proof question.
Prove that for any $n\ge1$, $\sum_{i=1}^n f_i^2=f_nf_{n+1}$, where $f_n$ is the $n$'th Fibonacci number.
I have the base case and the induction hypothesis, and I know what I need to prove (substitute $n+1$ in for $n$'s on both sides of the equation) If someone can just guide me in the right direction on where to go, using induction that would be helpful. Thank You
Try adding $f_{n+1}^2$ to both sides of the equation you have, and using the properties of the fibonacci numbers to simplify the right side. Let me know if you need any help.
More explicitly, if $\Sigma_{i=1}^n f_i^2=f_n f_{n+1}$, then
$f_{n+1}^2+\Sigma_{i=1}^n f_i^2=f_n f_{n+1}+f_{n+1}f_{n+1}=f_{n+1}(f_n+f_{n+1})$ See where that can get you.