I am trying to understand the proof that "For all $n\geq 2, F_n^2-F_{n+1}F_{n-1}=(-1)^{n-1}$.Where $F_n$ stands for the Fibonacci number at $n$. I got this proof from a book and here is the proof.
Fn+12 = Fn+1 (Fn+Fn-1) by the definition of Fn+1
= Fn+1Fn+ Fn-1Fn+1 + (Fn2 - Fn2)
=Fn2 + Fn+1Fn - (Fn2-Fn-1Fn+1)
=Fn(Fn+Fn+1)-(Fn2-Fn-1Fn+1)
=FnFn+2 - (Fn2 - Fn-1Fn+1) by definition of Fn+2
FnFn+2 -(-1)n-1 by the inductive assumption
Hence Fn+12 - FnFn+2 = (-1)n and the result follows.
I understand most of the proof I just don't understand the last part
FnFn+2 -(-1)n-1 by the inductive assumption
Hence Fn+12 - FnFn+2 = (-1)n and the result follows.
How does this part prove this thoerom? Arent'nt they just assuming that (Fn2 - Fn-1Fn+1) = (-1)n and not proving it?
I presume the base cases, $n=1$ and $n=2$ were shown by calculation. The idea of induction is to show that if it is true for $n$, it is true for $n+1$. Having shown the base cases, you assume it is true for $n$, and prove it is true for $n+1$. The inductive assumption is $F_n^2-F_{n+1}F_{n-1}=(-1)^{n-2}$ and we want to prove $F_{n+1}^2-F_{n+2}F_{n}=(-1)^{n-1}$ so we are allowed to use the former.
One of my favorites on this subject is Arturo Magidin's answer here