Fibonacci Number

Let's think geometrically.

By definition $f_{n+1}=f_n+f_{n-1}$

Let us draw a square of side-length $f_1=1$ first, then put another one but with side-length $f_2=1$ beisde the other one. So that they share a side. Now we can put a square of side $f_3=2$ right next to the other ones so that a border is shared and so on...

For example see here

Note now that, the area of the big rectangle is equal to the sum of the areas of the small squares inside. So, if the sides of the rectangle are $f_n$ and $f_{n+1}$ , it follows that $f_{n+1}\cdot{f_n}=f_n^2+f_{n-1}^2+...+f_1^2$

You may also try using induction