Imagine a bacterial population growing exponentially (doubling its size at each generation). The population starts at $t=0$ with $x_0=1$ cells, so $x_t=2^t$. Each generation cells copy themselves, having a probability $p$ per copy of generating a mutation. The question is: how many generations will it take on average to observe the first mutation?
Let's $Y$ be the number of generations to the first mutation, we can compute the expectation as:
$$ E(Y)=1·p+2·(1-p)(1-(1-p)^{2})+3·(1-p)^{3}(1-(1-p)^{4}) +... $$
$$ E(Y)=\sum_{t=1}^{k} t (1-p)^{2^{t-1}-1}(1-(1-p)^{2^{t-1}})$$
which looks like the expectation for the geometric distribution but with a exponentially increasing probability that at some generation $k=log_2(1/p)$ reaches unity.
I'd like to come up with a formula for this expectation, but I don't know even where to start (the common ratio trick doesn't seem possible). I wonder if this kind of series has a specific name that I can look up for (a related question has been asked here: Geometric distribution with unequal probabilities for trials)