Problem 5

Let

a_1,\ldots,a_8

be real numbers, not all zero, and let

c_n=a_1^n+\cdots+a_8^n

for

n=1,2,\ldots

. Suppose infinitely many terms

c_n

are zero. Determine all values of

n

for which

c_n=0

Step 7 of 7: State the complete condition

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\{a_1,\ldots,a_8\}=\{x_1,-x_1,\ldots,x_4,-x_4\}

Detailed analysis

Every nonzero magnitude group is balanced, and zeros may be paired with zeros. Hence the multiset is four opposite pairs. Conversely, each pair contributes $x_j^n+(-x_j)^n=0$ for odd $n$ , so infinitely many $c_n$ vanish.