Simplifying Fibonacci Identities

Question

This is very simple, but I am trying to simplify down Fibonacci sequences when doing induction problems, and I am very confused by this. For example, I was trying to simplify $F_n * F_{n+1} + F^2_{n+1})$ down to $F_{n+1} * F_{n+2}$

Answer 1

You just have to use the definition of the Fibonacci sequence:

$$F_{n+2}=F_{n}+F_{n+1}$$

Factoring $F_{n+1}$ we have,

$$F_{n+1}(F_{n}+F_{n+1})$$

Now it should be clear.

Answer 2

$F_nF_{n+1}+F_{n+1}^2=(F_n+F_{n+1})F_{n+1}=F_{n+2}F_{n+1}$