Complex Power series - figuring out the coefficient

Question

Having trouble figuring out what the coefficient of the series is for this question:

$$\sum_{n=0}^\infty \frac{z^{3n}}{2^n}$$

Once I get the coefficient I'm sure I'll be able to figure out the radius of convergence as the question asks

Answer 1

$$\sum_{n=0}^\infty \frac{z^{3n}}{2^n}$$ $$=\frac{z^0}{2^0}+\frac{z^3}{2^1}+\frac{z^6}{2^2}+\frac{z^9}{2^3}+\sum_{n=4}^\infty \frac{z^{3n}}{2^n}$$

Does that help ?

Answer 2

$a_n = \frac{1}{2^{n/3}}$ if $n=3k$ for any $k \in \Bbb{N}\cup \{0\}$ and $a_n=0$ in other case.