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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

1 Answers1

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$F_{n+1}=F_n+F_{n-1}$ by definition of the Fibonacci sequence.

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