Probability generating functions - please spot my mistake!

According to Zeno (495-435 BC) there was a race in which Achilles had to catch a tortoise having given it a start. Suppose that the race began with Achilles waiting on the starting line while the tortoise was given a start of r time units, where r is a positive integer. Suppose also

that, during each time unit, the tortoise either moves 1 metre with probability p and stays where

it is with probability q (p + q) = 1. **Write down the distribution of the distance travelled by the tortoise**

before it rst fails to move and find the probability generating function.

My answer: geometric, parameter p

Generating function: qs/(1-ps)

Once he has started, Achilles runs at a constant rate of 1 metre per time unit.

**Find the probability generating function for the distance from the start at which Achilles catches the tortoise and show that its expectation is rp/q**

My answer: Find where tortoise has not moved r times

Binomial distribution, parameter q

generating function: (p + qs)^n

I have put my first genearting function is this one, and differentuiated, and put s = 1 to give just n, so I mustve made a mistake

Many Many Thanks - I've been stuck on this for days!