Fibonacci Identity Proof for Weird Identity

Question

I found this identity that said: $F_{i+k}=F_{k-2}F_{i+1} + F_{k-1}F_{i+2}$. Is there any way to prove this? I haven't been able to make use of the typical Fibonacci substitutions to make any progress.

Answer 1

This does not hold in general. If we let $i=0$, then $$F_k=F_{k-2}\cdot 1+F_{k-1}\cdot 2$$

Edit: (Let us assume that, $F_0=0$, $F_1=1$ and $F_2=1$.) Use induction on $i$, for $i=0$ we get $$F_{k-2}F_{1+0}+F_{k-1}F_{2+0}=F_{k-2}F_{1}+F_{k-1}F_{2}=F_{k-2}+F_{k-1}=F_{k}=F_{k+0}.$$

Now suppose it holds for $n\leq i$, then

$$F_{k+i+1}=F_{k+i}+F_{k+i-1}=(F_{k-2}F_{i+1}+F_{k-1}F_{i+2})+(F_{k-2}F_{i}+F_{k-1}F_{i+1})=$$ $$(F_{k-2}F_{i+1}+F_{k-2}F_{i})+(F_{k-1}F_{i+2}+F_{k-1}F_{i+1})=F_{k-2}F_{i+2}+F_{k-1}F_{i+3}.$$