If $f_1, f_2, f_3,\ldots$ is the Fibonacci sequence proof $f_1^2 + f_2^ 2 + \cdots + f_n^2 = f_n f_{n+1}$.

Question

I'm assuming this is using strong induction/ regular induction. However, besides the "base case" I'm really confused with the inductive steps in my notes. The inductive steps in my notes use the alpha (golden ratio) but I'm confused on how that connects with the proof and how the it's used.

Answer 1

HINT:

If $\sum_{r=1}^m f^2_r=f_mf_{m+1}+1$,

$$\sum_{r=1}^{m+1} f^2_r=\sum_{r=1}^m f^2_r+f^2_{m+1}=f_mf_{m+1}+1+f^2_{m+1}=1+f_{m+1}(f_m+f_{m+1})$$

$$f_m+f_{m+1}=\text{?}$$