Complex Numbers Exercise If $a,b,c$ are complex numbers with $a+b+c=0$ and $\|a\|=\|b\|=\|c\| = r>0$ then prove that $$a^{2^n} + b^{2^n} + c^{2^n} = 0$$
Any ideas?
Thanks! 

Think of $a,b,c$ as vectors in the complex plane. Putting the tail of $a$ at zero, then putting the tail of $b$ to the tip of $a$, then putting the tail of $c$ to the tip of $b$, since $a+b+c=0$ it follows that the tip of $c$ is zero. Thus, the vectors form a triangle. This is an equilateral triangle of side length $r$. Now think about what happens geometrically when you raise a complex number to an integer power $k$. Then draw the triangle with side lengths $a^k,b^k,c^k$. 
