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 4 of 7: Balance the signs
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#{i:ai=R}=#{i:ai=−R}\#\{i:a_i=R\}=\#\{i:a_i=-R\}
Detailed analysis

If nn were even, every top term would be Rn>0R^n>0, impossible. Thus the selected large zero index is odd, and for odd nn the top sum is RnR^n times (number of +R+R minus number of −R-R). It vanishes exactly when the two counts are equal.