I have to prove that $$ F_{n}^{2} - F_{n+1}F_{n-1} = (-1)^{n} \;\;\;\;\;\;(n>1)$$
By induction, we can see this is true for $n=2$ (note that the sequence starts with $F_{0} = 1$). When proving the result for $n \ge 3$, the solution begins with $$F_{n}^{2} - F_{n+1}F_{n-1} = F_{n}^{2} - (F_{n}+F_{n-1})F_{n-1} = \cdot \cdot \cdot $$
Can someone please explain how we get that $$F_{n+1}F_{n-1} = (F_{n}+F_{n-1})F_{n-1} \;?$$
It is not immediately clear to me. If we multiply out the RHS it gives $$F_{n}F_{n-1} + F_{n-1}F_{n-1}$$ which I don't see how is equal to the LHS of the third equation I wrote. All help appreciated