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

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.

• I wonder why an edit was approved that had $f_n+1$ where $f_{n+1}$ belongs? $\qquad$ – Michael Hardy Mar 25 '16 at 15:41

HINT:

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

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

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

• You gave already a nice answer here. – Dietrich Burde Mar 25 '16 at 15:39
• @DietrichBurde, Thanks for pointing out. Was that inductive? – lab bhattacharjee Mar 25 '16 at 15:42