MathLabs

Problem 5

Let a1,…,a8a_1,\ldots,a_8 be real numbers, not all zero, and let cn=a1n+⋯+a8nc_n=a_1^n+\cdots+a_8^n for n=1,2,…n=1,2,\ldots. Suppose infinitely many terms cnc_n are zero. Determine all values of nn for which cn=0c_n=0.
Step 3 of 7: Force cancellation at the top
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∑∣ai∣=Rain=0\sum_{|a_i|=R}a_i^n=0
Detailed analysis

The top contribution is an integer multiple of RnR^n, while the remaining contribution has absolute value less than RnR^n. Since their sum is zero, the top contribution itself must be zero.