<s>[INST] How do I solve a problem with term $a^{n} + b^{n}$? 
Given two non-zero numbers $x$ and $y$ such that $x^{2} + xy + y^{2} = 0$.
Find the value of
$$\left(\frac{x}{x + y}\right)^{2013} + \left(\frac{y}{x + y}\right)^{2013}$$.

I found out that $(x + y)^2 = xy$ and I'm stuck at $\frac{x^{2013} + y^{2013}}{(x + y)^{2013}}$
Does anyone know how to solve this?
 [/INST] Since $y=x\exp\frac{\pm2\pi i}{3}$,$$\frac{x^n+y^n}{(x+y)^n}=\frac{1+\exp\frac{\pm2\pi i n}{3}}{(1+\exp\frac{\pm2\pi i}{3})^n}=\frac{2\exp\frac{\pm\pi i n}{3}\cos\frac{\pi n}{3}}{(2\exp\frac{\pm\pi i}{3}\cos\frac{\pi}{3})^n}=2\cos\frac{\pi n}{3}.$$In the case $n=2013$, this simplifies to $-2$ because $n/3$ is odd.
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