You're correct. Add up all the values you've gotten. This is a geometric series:
(1/6) * [ 1 + x + x^2 +x ^3 + ...]
where x = (5/6)(5/6).
The problem is following:
A and B takes turns throwing a dice. A begins. Who throws six first wins. What's the probability that A wins?
I tried to examine probabilities that A wins on different rounds. Something like this:
But I can't solve the general form (or something). I think it has something to do with binomial probability and geometric series... Any help is appreciated, thanks!