Fibonacci numbers identity - proof by induction

Question

$\displaystyle F_{k-1} F_{k+1} - F_k^2 = (-1)^k$

I have done the base step for $k=1$ and it works.

I realize we need to prove for $k+1$, so:

$$F_k F_{k+2} - F_{k+1}^2 = (-1)^{k+1}$$

Could someone please tell me how to actually continue proving this?

Answer 1

Write $F_{k+2}$ as $F_{k+1}+F_{k}$, you have:

$$F_k F_{k+2} - F_{k+1}^2=F_{k}(F_{k+1}+F_{k})-F_{k+1}^2=F_{k}F_{k+1}+F_{k}^2-F_{k+1}^2=F_{k+1}(F_{k}-F_{k+1})+F_{k}^2$$

Now $F_{k}-F_{k+1}=-F_{k-1}$, so:

$$F_k F_{k+2} - F_{k+1}^2=-F_{k+1}F_{k-1}+F_{k}^2=(-1)^{k} \cdot (-1)=(-1)^{k+1}$$