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 6 of 7: Identify all zero indices

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c_n>0\text{ for even }n,\quad c_n=0\text{ for odd }n

Detailed analysis

For even $n$ , every nonzero pair contributes $2x_j^n>0$ , so $c_n>0$ because not all numbers are zero. For odd $n$ , every opposite pair cancels. Therefore exactly the odd positive integers satisfy $c_n=0$ .