## Branching Process.

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

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

(i) Express $A(z)$ as a power series and hence find $P(N=6)$.
(ii) If $q_n = P(X_n = 0)$ for n = 0,1, ..., write down an equation relating $q_{n+1}$ and $q_n$. Hence, or otherwise, evaluate $q_n$ for n = 0,1,2.
(iii) Find the extinction probability $q = lim_{n \rightarrow \infty} q_n$.

Thank youu!