Erdős problem 974

Let $z_1,\ldots,z_n\in \mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence of $$s_k=\sum_{1\leq i\leq n}z_i^k$$ contains infinitely many $(n-1)$-tuples of consecutive values of $s_k$ which are all $0$. Then (essentially) $$z_j=e(j/n),$$ where $e(x)=e^{2\pi ix}$.