Branching process, inductive proof

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Homework Statement



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

Show by induction that:

[tex]G_n(s)=\frac{1-2^n-2(1-2^{n-1})s}{1-2^{n+1}-2(1-2^n)s}[/tex]

2. The attempt at a solution

I can see that the distribution is geometric so:

[tex]G(s)=\frac{p}{1-qs}=\frac{1}{3-2s}[/tex]

I assume I have to show that:

[tex]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}}[/tex]

equals:

[tex]\frac{1-2^{n+1}-2(1-2^{n})s}{1-2^{n+2}-2(1-2^{n+1})s}[/tex]

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?
 

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