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(A) If $n$ is not a multiple of $3$,then $n = 3k + 1$ or $n = 3k + 2$,where $k \in \mathbb{Z}$.Case $(I)$: When $n = 3k + 1$.$\omega^n + \omega^{2n} =

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(A) If $n$ is not a multiple of $3$,then $n = 3k + 1$ or $n = 3k + 2$,where $k \in \mathbb{Z}$.Case $(I)$: When $n = 3k + 1$.$\omega^n + \omega^{2n} = \omega^{3k+1} + \omega^{6k+2} = (\omega^3)^k \cdot \omega + (\omega^3)^{2k} \cdot \omega^2 = 1^k \cdot \omega + 1^{2k} \cdot \omega^2 = \omega + \omega^2$.Since $1 + \omega + \omega^2 = 0$,we have $\omega + \omega^2 = -1$.Case $(II)$: When $n = 3k + 2$.$\omega^n + \omega^{2n} = \omega^{3k+2} + \omega^{6k+4} = (\omega^3)^k \cdot \omega^2 + (\omega^3)^{2k} \cdot \omega^4 = 1^k \cdot \omega^2 + 1^{2k} \cdot \omega = \omega^2 + \omega = -1$.In both cases,$\omega^n + \omega^{2n} = -1$ for all $n$ that are not multiples of $3$.

$n$ is a positive integer and not a multiple of $3$. If $\omega$ is a non-real cube root of unity,then $\omega^n + \omega^{2n}$ is equal to

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