Fibonacci numbers $F_{n+3} + F_{n} = 2F_{n+2}$

Question

Prove $F_{n+3} + F_{n} = 2F_{n+2}$ for any positive integer n.

So What I did was this:

fn+ fn+1 = fn+2

fn + fn+1 = fn+2 => fn+2 -fn+1 fn+1 + fn+2 = fn+3

then I subsituted into equation in question and got:

fn+2-fn+fn+2=fn+3=fn+3 + fn=2n+2

So that makes sense. But I am having trouble grasping why this proves to be true for any integer.

Answer 1

For all $n$, $f_n+f_{n+1}=f_{n+2}$ (Fibonacci numbers). For this reason, $f_{n+1}+f_{n+2}=f_{n+3}.$

So, $f_n+(f_{n+1}+f_{n+2})=f_n+f_{n+3}$, and $(f_n+f_{n+1})+f_{n+2}=2f_{n+2}.$