Fibonacci trick and proving it. I am trying to learn Fibonacci tricks and I have one that I can not prove. I know it works because Ive tried it multiple times but I have not a clue how to prove. Here it is:
f(0)^2 + f(1)^2 + f(2)^2 + f(3)^2 = f(3)f(3+1)
  0    +   1    +   1    +   4    =   2  *  3
            = 6                          =6 

Is there a way to prove this?

Basis : ok
Inductive step : Assume $∑_{i=0}^{n}f(i)^2=f(n)f(n+1)$.
$$∑_{i=0}^{n+1}f(i)^2=f(n)f(n+1)+f(n+1)^2$$$$∑_{i=0}^{n+1}f(i)^2=f(n+1)(f(n)+f(n+1))$$$$∑_{i=0}^{n+1}f(i)^2=f(n+1)f(n+2)$$
There is a geometric interpretation:

