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$\displaystyle F_{k-1} F_{k+1} - F_k^2 = (-1)^k$

I have done the base step for $k=1$ and it works.

I realize we need to prove for $k+1$, so:

$$F_k F_{k+2} - F_{k+1}^2 = (-1)^{k+1}$$

Could someone please tell me how to actually continue proving this?

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Have you expanded all terms using the definition of $F_k$ to see what happens? –  abiessu Jul 9 at 18:19

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up vote 2 down vote accepted

Write $F_{k+2}$ as $F_{k+1}+F_{k}$, you have:

$$F_k F_{k+2} - F_{k+1}^2=F_{k}(F_{k+1}+F_{k})-F_{k+1}^2=F_{k}F_{k+1}+F_{k}^2-F_{k+1}^2=F_{k+1}(F_{k}-F_{k+1})+F_{k}^2$$

Now $F_{k}-F_{k+1}=-F_{k-1}$, so:

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

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