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If $\omega$ is an $n^{\text {th }}$ root of unity, other than unity, then the value of $1+\omega+\omega^2+\ldots+\omega^{n-1}$ is
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$0$
$1+\omega+\omega^2+\ldots . .+\omega^{n-1}=\left(\frac{\omega^n-1}{\omega-1}\right)=0,\left(\omega^n=1\right)$
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