Sum of roots of unity

Geometric series. Since $\omega^n=1$ and $\omega\neq1$ (for $n\ge2$),
$$\sum_{k=0}^{n-1}\omega^k=\frac{1-\omega^n}{1-\omega}=\frac{1-1}{1-\omega}=0.$$

Conclusion. The sum of all $n$-th roots of unity is zero.

Interpretation. The centroid of the regular $n$-gon inscribed in the unit circle is at the origin.