Think I might need to refresh myself in power series before doing this one. Anyway, how should I proceed with this question?

Consider the Branching Process {$\displaystyle X_n, n = 0, 1, 2, 3, ...$} where $\displaystyle X_n$ is the population size of the nth generation. Assume $\displaystyle P(X_o = 1) = 1$ and that the pgf of the common offspring distribution N is

$\displaystyle A(z) = \frac{1}{3 - 2z}$

(i) Express $\displaystyle A(z)$ as a power series and hence find $\displaystyle P(N=6)$.

(ii) If $\displaystyle q_n = P(X_n = 0)$ for n = 0,1, ..., write down an equation relating $\displaystyle q_{n+1}$ and $\displaystyle q_n$. Hence, or otherwise, evaluate $\displaystyle q_n$ for n = 0,1,2.

(iii) Find the extinction probability $\displaystyle q = lim_{n \rightarrow \infty} q_n$.

Thank youu!