For the Fibonacci numbers, show for all $n$: $F_1^2+F_2^2+\dots+F_n^2=F_nF_{n+1}$ The definition of a Fibonacci number is as follows:
$$F_0=0\\F_1=1\\F_n=F_{n-1}+F_{n-2}\text{ for } n\geq 2$$ 
Prove the given property of the Fibonacci numbers for all n greater than or equal to 1.
$$F_1^2+F_2^2+\dots+F_n^2=F_nF_{n+1}$$
I am pretty sure I should use weak induction to solve this. My professor got me used to solving it in the following format, which I would like to use because it help me map everything out...
This is what I have so far:
Base Case: Solve for $F_0$ and $F_1$ for the following function: $F_nF_{n+1}$.
Inductive Hypothesis: 
What I need to show: I need to show $F_{n+1}F_{n+1+1}$ will satisfy the given property.
Proof Proper:  (didn't get to it yet)
Any intro. tips and pointers?

The inductive assumption for $n$ is 
$$F(1)^2+\ldots+F(n)^2=F(n)F(n+1)$$.
Using this, the $n+1$ case is:
$$\begin{align*} F(1)^2+\ldots +F(n)^2+F(n+1)^2 &=F(n)F(n+1)+F(n+1)^2 \\
&=F(n+1)(F(n)+F(n+1)) \\
&=F(n+1)F(n+2)\end{align*} $$
