Question on Discrete Parameter Markov Chains

[AD]

I am required to find a formula expressing the probability of return to some state in a Markov chain at time n in terms of the probability of return to that state at time n - k and the probability of first return at time k. I cannot find this in my notes, and I have tried looking at several online resources. Can anyone help me?

 

[AD]

It is no longer necessary for you to answer this question as I have just discovered the answer elsewhere.

 

[Hurkyl]

You could just derive it. How many ways can you return at time n?

One way is to have time n be the first time you return.
A second way is to return at time 1, and then have n be the next time you return.
Yet another way is to return at time 2, and then have n be the next time you return...