Sersa * Does there exist a positive integer $n$, such that the complex number $z=\left(\frac{3-\sqrt{3} \mathrm{i}}{2}\right)^{n}$ is a pure imaginary number?
Parse: Because $z=\left(\frac{3-\sqrt{3} \mathrm{i}}{2}\right)^{n}=\left[\sqrt{3} \mathrm{i}\left(-\frac{1}{2}-\frac{\sqrt{3}}{2} \mathrm{i}\right)\right]^{n}=(\sqrt{3} \mathrm{i} \cdot \bar{\omega})^{n}=(\sqrt{3})^{n} \cdot \mathrm{i}^{n} \cdot(\bar{\omega})^{n}$, hence when $n=3 k\left(k \in \mathbf{N}^{*}\right)$, the complex number $z= \pm(\sqrt{3})^{n} \mathrm{i}$ is a pure imaginary number.