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## Homework Statement

Consider a branching process with branching probabilities given by P

_{0}=1/2 and P

_{j}=1/3[tex]^{j}[/tex] for j [tex]\geq[/tex] 1

Find the probability generating function: [tex]\sum^{\infty}_{n=0} p_{n}x^{n}[/tex]

## The Attempt at a Solution

Now, the answer is supposed to be G(x) = (x+3)/(2(3-x)), but I really have no idea how to get this answer. I've checked it up on the internet and haven't found anything that looks even vaguely helpful, and since I'm only meeting my tutor later in the week I'm pretty much stuck on my own for this one. Is there some sort of formula you can use to find the probability generating function? If possible, could you please show me how to get that answer from the given data?

Thanks in advance.