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 $S_5 + S_{11}+S_{21}=S_8.$

#Algebra #FibonacciNumbers #SystemOfNonLinearEquations #GCDC

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Comments

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 $(1^5 + 1^5 + (-1)^5) + (1^{11} + 1^{11} + (-1)^{11}) + (1^{21} + 1^{21} + (-1)^{21}) = 1^8 + 1^8 + (-1)^8$ $1+1+1=3$

Guilherme Dela Corte - 6 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$