Question 5
Consider the sequence $(c_n)$:
$$\begin{aligned}c_1&=a_1+a_2+\cdots+a_8,\\c_2&=a_1^2+a_2^2+\cdots+a_8^2,\\&\cdots\\c_n&=a_1^n+a_2^n+\cdots+a_8^n,\\&\cdots\end{aligned}$$
where $a_1,a_2,\ldots,a_8$ are real numbers, not all equal to zero. Given that among the numbers of the sequence $(c_n)$ there are infinitely many equal to zero, determine all the values of $n$ for which $c_n=0$.
$$\begin{aligned}c_1&=a_1+a_2+\cdots+a_8,\\c_2&=a_1^2+a_2^2+\cdots+a_8^2,\\&\cdots\\c_n&=a_1^n+a_2^n+\cdots+a_8^n,\\&\cdots\end{aligned}$$
where $a_1,a_2,\ldots,a_8$ are real numbers, not all equal to zero. Given that among the numbers of the sequence $(c_n)$ there are infinitely many equal to zero, determine all the values of $n$ for which $c_n=0$.