Induction proof $F(n)^2 = F(n-1)F(n+1)+(-1)^{n-1}$ for n $\ge$ 2 where n is the Fibonacci sequence Prove that $F{_n}^2 = F_{n-1}F_{n+1}+(-1)^{n-1}$ for n $\ge$ 2 where n is the Fibonacci sequence
F(2)=1, F(3)=2, F(4)=3, F(5)=5, F(6)=8 and so on.
Initial case n = 2: 
$$F(2)=1*2+-1=1$$  It is true.
Let k = n $\ge$ 2
To show it is true for k+1
How to prove this?

\begin{align}
F_n\cdot F_{n+2} + (-1)^n &= F_n\cdot\left(F_{n+1} + F_n\right) + (-1)^n 
\\&= F_n\cdot F_{n+1} + F^2_n + (-1)^n 
\\&= F_n\cdot\left(F_n + F_{n-1}\right) + F^2_n + (-1)^n 
\\&= F^2_n + F_n\cdot F_{n-1} + F^2_n + (-1)^n 
\\&= F^2_n + F_n\cdot F_{n-1} + \left(F^2_n - (-1)^{n-1}\right) 
\\&= F^2_n + F_n\cdot F_{n-1} + F_{n-1}\cdot F_{n+1} 
\\&= F^2_n + F_n\cdot F_{n-1} + F_{n-1}\cdot \left(F_n + F_{n-1}\right) 
\\&= F^2_n + F_n\cdot F_{n-1} + F_{n-1}\cdot F_n + F^2_{n-1} 
\\&= F^2_n + 2\cdot F_n\cdot F_{n-1} + F^2_{n-1} 
= \left(F_n + F_{n-1}\right)^2 = F^2_{n+1}
\end{align}
