Q:

$-\frac{(-1)^n \left(\phi ^2+1\right)^2}{5 \phi ^2}=-e^{i \pi n}$

Can someone explain how: $$-\frac{(-1)^n \left(\phi ^2+1\right)^2}{5 \phi ^2}=-e^{i \pi n}$$, where $\phi$ is the golden ratio

A:

Not-so-subtle hint: $\phi=\frac{1+\sqrt{5}}{2}$ and $\phi^2=\phi+1$, so that

$$(\phi^2+1)^2 =(\phi+2)^2 =\left(\frac{1+\sqrt{5}}{2} +2\right)^2 =\left(\frac{5+\sqrt{5}}{2}\right)^2 =\left(\sqrt{5} \cdot \frac{1+\sqrt{5}}{2}\right)^2 =5\phi^2.$$

