## Summation Equation, Trying to solve this recurrence forumla.

Hello. I've searched around a bit for a math forum where I could get help with this and this seems like the one I found where I could get some help with this. I was posed the following problem. Now I must admit it is over my head (as is most of the math on this forum) I was hoping that someone here could help give me an answer for this person. Or if not an answer, a reason why his probem makes no sense.

 I want to predict the number of cups to be hit in beer pong each round, based upon shot percentage. It's not just number of shots * percentage, because if you make 2 shots in a row, you get an extra turn. Even your extra turns can get extra turns.. Therefore, Number of cups per round is the recursive formula C(t) = C(a^2 * t) + 2at where 0 < a < 1 for accuracy So if you want to find the number of cups in 1 round, calculate c(1) This can actually be reduced to the summation equation for i = 0 to infinity -> 2a^(2i+1) How do I solve this summation and get a formula? I'd really appreciate it, Thanks.

 Recognitions: Homework Help Science Advisor You either miss your first shot, OR make your first and miss the next, OR make your first two and miss the next OR ... OR make the first n and miss the next OR ... The probability that you make the first n and miss the next is an(1-a). So what you want is the sum of (getting n cups) x (probability of getting n cups) for all n, i.e.: $$\sum _{n=0}^{\infty}na^n(1-a) = (1-a)\sum _{n=0}^{\infty}na^n = (1-a)\sum _{n=1}^{\infty}na^n = a(1-a)\sum_{n=1}^{\infty}na^{n-1} = a(1-a)f'(a)$$ where $$f(x) = \sum_{n=1}^{\infty}x^n = \sum_{n=0}^{\infty}x^n - 1 = \frac{1}{1-x} - 1 = \frac{x}{1-x}$$ So $$f'(x) = \frac{(1-x) - x(-1)}{(1-x)^2} = \frac{1}{(1-x)^2}$$ So $$f'(a) = \frac{1}{(1-a)^2}$$ Finally, the desired number is: $$a(1-a)\frac{1}{(1-a)^2} = \frac{a}{1-a}$$
 thank you very much