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 \<br />\Leftrightarrow \sqrt{5}k = \alpha^n - \beta^n \<br />\Leftrightarrow 5k^2 = (\alpha^n - \beta^n)^2 \<br />\Leftrightarrow 5k^2 = \alpha^{2n} + \beta^{2n} - 2 (\alpha \beta) ^n \<br />\Leftrightarrow 5k^2 = \alpha^{2n} + \beta^{2n} - 2(-1)^n \<br />\Leftrightarrow 5k^2 + 4(-1)^n = \alpha^{2n} + \beta^{2n} - 2(-1)^n + 4(-1)^n\<br />\Leftrightarrow 5k^2 + 4(-1)^n = \alpha^{2n} + \beta^{2n} + 2(-1)^n \<br />\Leftrightarrow 5k^2 + 4(-1)^n = \alpha^{2n} + \beta^{2n} + 2 (\alpha \beta)^n \<br />\Leftrightarrow 5k^2 + 4(-1)^n = (\alpha^n + \beta^n)^2 \hspace{50pt}(1)\<br />)