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# Fibonacci Strike

Let $$a,b,c$$ be complex numbers, and $$S_n = a^n+b^n+c^n$$ be the sum of their $$n$$-th powers.

If $S_1=1, \; S_1=1, \; S_2=3, \; S_3=1$ Show that $S_5 + S_{11}+S_{21}=S_8.$

Note by Guilherme Dela Corte
3 years, 1 month ago

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From Newton Sums, one can find that $$a+b+c=1, ab+ac+bc=-1, abc=-1$$. The polynomial $$p(x)$$ of roots $$a,b,c$$ can thus be written as $$p(x)=x^3-x^2-x+1$$. This leads to an adequate way to show that $$1$$ is a double root and $$-1$$ is a simple root, meaning $$a=b=1, \; c=-1$$, for instance.

It is easy to see that our desired equation yields, since $(1^5 + 1^5 + (-1)^5) + (1^{11} + 1^{11} + (-1)^{11}) + (1^{21} + 1^{21} + (-1)^{21}) = 1^8 + 1^8 + (-1)^8$ $1+1+1=3$

- 3 years, 1 month ago