If is a root of the equation x^2+x+1=0 and _k=1^n( ^k+1 ^k)^2=20, then n is equal to _____…

Question

If $\alpha $ is a root of the equation ${x}^{2}+x+1=0$ and $\sum _{k=1}^{n}{({\alpha }^{k}+\frac{1}{{\alpha }^{k}})}^{2}=20$, then n is equal to \_\_\_\_\_\_\_\_\_.

Answer: 11

Accepted Answer

$\alpha$ is a non-real cube root of unity, $\alpha=\omega$ or $\omega^2$. For each $k$:
- If $3\mid k$: $\alpha^k=1$, so $\left(\alpha^k+\dfrac1{\alpha^k}\right)^2=(1+1)^2=4$.
- If $3\nmid k$: $\alpha^k+\dfrac1{\alpha^k}=\omega^k+\omega^{-k}=-1$ (the two non-real cube roots sum to $-1$), so the square is $1$.

For $n=11$: multiples of 3 up to 11 are $3,6,9$ (three of them, each contributing 4), and the remaining $8$ values contribute $1$ each:
$$\sum_{k=1}^{11}\left(\alpha^k+\frac1{\alpha^k}\right)^2=3(4)+8(1)=12+8=20.$$

This matches, so $n=11$.

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