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**FGT12** Let $\displaystyle N_j$ be the size of the jth generation of a branching process with $\displaystyle N_0 = 1$. Suppose the number of offspring produced by an individual has mean $\displaystyle \mu$ and variance $\displaystyle {\sigma}^2$. Let $\displaystyle \alpha_j = \frac{Var(N_j)}{{{\sigma}^2}{\mu}^{j-1}}$

Show that $\displaystyle \alpha_j = \alpha_{j-1}+\mu^{j-1}$

I get to:

$\displaystyle \frac{Var(N_{j-1})}{\sigma^2\mu^{j-2}}+\mu^{j-1}$

$\displaystyle =\frac{Var(N_{j-1})+{\sigma^2\mu^{2j-3}}}{{\sigma^2\mu^{j-2}}}$

However as I am not seeing the expression for $\displaystyle Var(N_j)$ which enables you to solve this question