I am to prove by induction for the fibonacci numbers $F_1 = 1$, $F_2 = 1$, $F_n = F_{n-1}+F_{n-2}$ the following $P(n)$ is true: $$\left(\forall n \in \mathbb{N}^{+}\right)\left[\left(F_{n+1}\right)^{2}-\left(F_{n+2} \cdot F_{n}\right)=(-1)^{n}\right]$$ So far, I have written and expanded out $P(n+1)$, but I cannot find where I am to use the inductive hypothesis of $P(n)$ to reach any conclusion. I start with $$P(n+1) = (F_{n+1})^2+2F_{n+1}F_n+(F_{n})^2-F_{n+2}F_n-F_{n+2}F_{n-1}-F_{n+1}F_n-F_{n+1}F_{n-1}$$ and this eventually reduces to $$F_{n+1}F_n-F_{n+2}F_{n+1}$$ with my inductive hypothesis applying for $P(n)$ and $P(n-1)$ to make some terms go away. However, as I continue with this method, I make no progress towards my goal of simplifying to $(-1)^n$. How can I make progress without simplifying this endlessly?
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2You probably made a mistake during simplification, since $F_{n+1}F_n - F_{n+2}F_{n+1} = F_{n+1} (-F_{n+1}) = -F_{n+1}^2$ – player3236 Oct 30 '20 at 03:22
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1See Cassini's Identity on ProofWiki. – player3236 Oct 30 '20 at 03:34
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Wow, there really is a name for everything. Looks like my problem was I was simply uncreative. Thanks so much! – sucks-at-javascript Oct 30 '20 at 03:58