In a branching process every individual has probability $p_{k}$ of producing exactly $k$ offspring, $k=0,1, \ldots$, and the individuals of each generation produce offspring independently of each other and of individuals in preceding generations. Let $X_{n}$ represent the size of the $n$th generation. Assume that $X_{0}=1$ and $p_{0}>0$ and let $F_{n}(s)$ be the generating function of $X_{n}$. Thus

$$F_{1}(s)=\mathbb{E} s^{X_{1}}=\sum_{k=0}^{\infty} p_{k} s^{k},|s| \leqslant 1$$

(a) Prove that

$$F_{n+1}(s)=F_{n}\left(F_{1}(s)\right)$$

(b) State a result in terms of $F_{1}(s)$ about the probability of eventual extinction. [No proofs are required.]

(c) Suppose the probability that an individual leaves $k$ descendants in the next generation is $p_{k}=1 / 2^{k+1}$, for $k \geqslant 0$. Show from the result you state in (b) that extinction is certain. Prove further that in this case

$$F_{n}(s)=\frac{n-(n-1) s}{(n+1)-n s}, \quad n \geqslant 1$$

and deduce the probability that the $n$th generation is empty.