Suppose one plays the following game: Start with $n$ rounds. Each round, flip a weighted coin with probability $p$ of coming up heads. Each time we obtain heads, we get $m$ additional rounds added to our current total. We play until we run out of rounds.
The expected number of rounds played is:
$$
E[\# Rounds]= n + nmp + n(mp)^2 + n(mp)^3 +\dots 
$$
My question is how does one derive this expectation formula?
(This is a question for work, not school).