For a Galton-Watson process, I've shown that $\frac{Z_n}{\mu^n}$ is a martingale i.e. $E[\frac{Z_{n+1}}{\mu^{n+1}}|\mathcal{F}_n]=\frac{Z_n}{\mu^n}$.
However, I want to show that, for $n>m$, $$E[Z_{n}|Z_m]=\mu^{n-m}Z_m.$$ But I don't understand how to compute the right-hand side using the knowledge that $\frac{Z_n}{\mu^n}$ is a martingale. I have been messing around with it for a little while, but I can't reach it.