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.