Fibonacci, prove that $F_{n}\cdot F_{n+2}-({F_{n+1}})^2=(-1)^n$ with induction

Question

I need to prove by induction that:

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

I did the following:

• Check if the statement holds for $n=1$:

$$1\cdot 3-(2)^2=(-1)^1$$

• Check if the statement holds for $n=p+1$:

$$F_{p+1}\cdot F_{p+3}-({F_{p+2}})^2=(-1)^{p+1}$$

I am having some troubles with this. I don't know how to continue.