Solution
= Solution
Given $X_n=k$, the next size is a sum of $k$ independent reproduction counts. Multiplication of their <probability generating functions> gives $\mathbb E[s^{X_{n+1}}\mid X_n=k]=G(s)^k$. Averaging this <conditional expectation> proves
$$
\boxed{G_{n+1}(s)=\mathbb E[G(s)^{X_n}]=G_n(G(s)).}
$$
With one initial individual, $G_0(s)=s$ and $G_1(s)=G(s)$, so $G_n$ is the $n$-fold composition of $G$ with itself.