= Solution
Differentiate the <probability generating function> equation:
$$
G_N'(s)=pG_N(s)^2+2psG_N(s)G_N'(s).
$$
At $s=1$, $G_N(1)=1$, giving
$$
\boxed{\mathbb E[N]=G_N'(1)=\frac p{1-2p}.}
$$
The <derivative> is finite because $p<1/2$ and the explicit square root has no singularity at $s=1$. Alternatively, the expected population in generation $t$ is $(2p)^t$, so the expected number of splits in that generation is $p(2p)^t$. Summing this nonnegative <geometric series> gives the same answer without initially assuming a finite <expected value>.
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