This isn't really the original Achilles & tortoise problem but it uses the characters:
Suppose that the race began with Achilles waiting on the starting line while the tortoise was given a start of t time units, where t is a positive integer. Suppose also that, during each time unit, the tortoise either moves 1 metre with probability p or stays where it is with probability 1-p. Write down the distribution of the distance travelled by the tortoise before it fails to move and find the probability generation function.
I could do this part because it's the geometric distribution, and the pgf is (1-p)/(1-sp).
Once he has started, Achilles runs at a constant rate of 1m per time unit. Find the pgf for the distance from the start at which Achilles catches the tortoise and show that its expectation is tp/(1-p).
I'm really stuck on this part.. Suppose the tortoise travels D metres before Achilles starts. Achilles will need to travel each time the tortoise travels, plus all the times the tortoise fails to travel, and there need to be D of these, so we need to find the amount of time it takes the tortoise to fail D times? It looks like a sum of I.I.D. random variables but I'm not sure how to go about it. Please help?