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?

Let $a_n = \left(\dfrac{x}{x + y}\right)^{n} + \left(\dfrac{y}{x + y}\right)^{n}$.
Since
$$
\dfrac{x}{x + y}+\dfrac{y}{x + y}=1, \quad
\dfrac{x}{x + y}\cdot\dfrac{y}{x + y}=1
$$
they are the roots of $t^2=t-1$ and so
we get
$$
a_{n+2} = a_{n+1}-a_n,
\quad
a_0=2,
\quad a_1=1
$$
This sequence is periodic of period $6$:
$$
2,1,-1,-2,-1,1,
2,1,-1,-2,-1,1,
\dots
$$
Thus, $a_{2013} = a_{2013 \bmod 6} = a_3 = -2$.
