Let X1, X2, .... be a sequence of iid random variables with PMF:

P(X=x)= [a(1-p)^x]/x, x=1,2,3...., with 1>1-p>0 and a=-1/log(p)

If N is independent of the Xi and has a Poisson distribution with parameter u (au is an integer) then show that S, the sum from i=1 to N of the Xi has a Negative Binomial distribution with PGF:

Gs(S)=[p/1-s(1-p)]^au

Any hints how to do this?

I know that Gs(S)=GN(Gx(s))

I have tried this many times by trying first to calculate Gx(s), with the thought that I should get something like the PGF of a geometric but am getting nowhere.

Can anyone hint of how to get started etc