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 5 of 7: Remove the largest group
Previous stepNext step
{ai:∣ai∣=R}={+R,−R} in balanced pairs\{a_i:|a_i|=R\}=\{+R,-R\}\text{ in balanced pairs}
Detailed analysis

Remove the balanced +R+R and −R-R terms. Their odd powers cancel identically, so the remaining numbers still have infinitely many zero odd power sums. Apply the same argument to the next magnitude group.