International Mathematical Olympiad 1967 Problem 5

Consider the sequence $\{c_n\}$, where $$\begin{aligned} c_1 &= a_1 + a_2 + \cdots + a_8 \\ c_2 &= a_1^2 + a_2^2 + \cdots + a_8^2 \\ &\cdots \\ c_n &= a_1^n + a_2^n + \cdots + a_8^n \\ &\cdots \end{aligned}$$ in which $a_1, a_2, \ldots, a_8$ are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence $\{c_n\}$ are equal to zero. Find all natural numbers $n$ for which $c_n = 0$.