I saw this step while reading up on branching proccesses. $X_i$ is the size of the in the $i$th generation and $\mu$ is the mean number of offspring per individual.
$E[X_n|X_{n-1}]$
$\mu E[X_{n-1}]$
So I'll try and reverse the steps...
$\mu E[X_{n-1}]$
$E[\mu \cdot X_{n-1}]$
$E[E[X] \cdot X_{n-1}]$
But I can't see how I can go from there to
$E[X_n|X_{n-1}]$
Is that some standard identity for expected values that I haven't seen?