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This looks like a straightforward recurrence, but I have an impression I made a mistake somewhere. In this equation $G_n$ is a random variable

$ G_n=\left\{ \begin{array}{c c} 0 & 1-p_n \\ a_n & p_n \\ \end{array} \right. $

Obviously $\mathbf{E}G_n=a_n p_n$ and border condition is $A_1=1$. So the obvious solution to the problem seems to be

$ A_{n+1}=1+\sum_{j=2}^{n}a_j p_j $

I find this solution way too simple. Do I have to use a generating function somewhere? I had this article in mind when thinking of this problem.

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    Apart from a typo in the index of summation, the calculation in the post is fine. The answer *is* simple, there is no need to complicate it. There are enough hard problems.2011-10-04

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