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 1 of 7: Order the magnitudes
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∣a1∣≥∣a2∣≥⋯≥∣a8∣|a_1|\ge|a_2|\ge\cdots\ge|a_8|
Detailed analysis

Reindex so that the absolute values are nonincreasing. Let the largest nonzero magnitude be RR, occurring rr times; all other magnitudes are strictly smaller.