Solution to If Fn is the nth Fibonacci number, prove that Fn+1Fn−1 − F2n = (−1)n. - Sikademy

Nov. 28, 2020

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Archangel Macsika

If Fn is the nth Fibonacci number, prove that Fn+1Fn−1 − F2n = (−1)n.

Solution to If Fn is the nth Fibonacci number, prove that Fn+1Fn-1 − Fn2 = (−1)n.

(\frac{\alpha^{n+1} - \beta^{n+1} }{\sqrt5})(\frac{\alpha^{n-1} - \beta^{n-1} }{\sqrt5})(\frac{\alpha^{n} - \beta^{n} }{\sqrt5})^2

and hence is equal to

\frac{2(\alpha\beta)^n-\alpha^{n+1}\beta^{n-1}-\alpha^{n-1}\beta^{n+1}}{5}.

Since αβ = −1, this is equal to

(-1)^n(\frac{2-\alpha\beta^{-1}-\alpha^{-1}\beta }{5})=(-1)^n(\frac{2+\alpha^2+\beta^2}{5}) =(-1)^n.