2. Find the smallest positive integer $n$, such that
$$
I=\left(\frac{1}{2}+\frac{1}{2 \sqrt{3}} \mathrm{i}\right)^{n}
$$

is a pure imaginary number, and find $I$.
Answer: $n=3, I=\frac{\sqrt{3}}{9} \mathrm{i}$.