3. If $x$ is a complex number satisfying $x^{2}+x+1=0$, what is the value of $x^{49}+x^{50}+x^{51}+$ $x^{52}+x^{53}$ ?
(A) -1
(B) $-\frac{1}{2}$
(C) 0
(D) $\frac{1}{2}$
(E) 1
3. Answer: (A)
Note that $x^{2}+x+1=\frac{x^{3}-1}{x-1}$, so $x^{2}+x+1=0$ implies that $x^{3}=1$ and $x \neq 1$. Now
$$
\begin{aligned}
x^{49}+x^{50}+x^{51}+x^{52}+x^{53} & =x^{49}\left(1+x+x^{2}\right)+x^{51}\left(x+x^{2}\right) \\
& =x^{49} \times 0+\left(x^{3}\right)^{17}(-1) \\
& =1^{17} \times(-1)=-1 .
\end{aligned}
$$