= Solution
During the epoch with $j$ ancestral lineages, <Kingman's coalescent> has waiting time $T_j\sim\operatorname{Exp}(\binom j2)$. In the stated time units, each lineage accumulates <mutations> at rate $Nu=\theta/2$, so the <mutation count on a neutral coalescent tree> decomposes as
$$
S=\sum_{j=2}^nY_j,\qquad Y_j\mid T_j\sim\operatorname{Poisson}\left(\frac{j\theta T_j}{2}\right).
$$
The <law of total expectation> and the mean of an <exponential distribution> give $\mathbb EY_j=(j\theta/2)/\binom j2=\theta/(j-1)$. Therefore
$$
\boxed{\mathbb ES=\theta H_{n-1},\qquad H_{n-1}=\sum_{i=1}^{n-1}\frac1i.}
$$
This calculation counts mutation events. Equating the count with observed <segregating sites> additionally uses the <infinite sites mutation model>.
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