# Branching process, inductive proof

1. Apr 17, 2012

### spitz

1. The problem statement, all variables and given/known data

Assume that the the offspring distribution is $P(Y=y)=\left(\frac{1}{2}\right)^y\frac{1}{3}$
$y=0,1,2,\ldots$

Show by induction that:

$$G_n(s)=\frac{1-2^n-2(1-2^{n-1})s}{1-2^{n+1}-2(1-2^n)s}$$

2. The attempt at a solution

I can see that the distribution is geometric so:

$$G(s)=\frac{p}{1-qs}=\frac{1}{3-2s}$$

I assume I have to show that:

$$G_{n+1}(s)=\frac{1-2^n-2(1-2^{n-1})\frac{1}{3-2s}}{1-2^{n+1}-2(1-2^n)\frac{1}{3-2s}}$$

equals:

$$\frac{1-2^{n+1}-2(1-2^{n})s}{1-2^{n+2}-2(1-2^{n+1})s}$$

The thing is, this seems like kind of a tedious question considering the amount of marks I'll get for it on my exam. Am I missing something here? Is there a "quick" way to do this?