Birth and Death Processes / First-order difference equation question

Hi Everyone,

I have become stuck on the following problem. Here is the statement:

In a simple birth and death process, the birth rates are

http://www.codecogs.com/gif.latex?p_%7Bi%7D=bi and the death rates are http://www.codecogs.com/gif.latex?q_%7Bi%7D=di for http://www.codecogs.com/gif.latex?i=1,%5Cdots,N-1. The birth and death rates are 0 elsewhere. The parameters http://www.codecogs.com/gif.latex?b and http://www.codecogs.com/gif.latex?d are positive and satisfy http://www.codecogs.com/gif.latex?%28b+d%29N%5Cleq1. Let http://www.codecogs.com/gif.latex?%5...#91;X_%7Bn%7D] . Show that http://www.codecogs.com/gif.latex?%5Cmu_%7Bn%7D satisfies the following first-order difference equation:

http://www.codecogs.com/gif.latex?%5...7Bn%7D=N%29%29.

I am thinking that I can use some sort of recurrence relation like http://www.codecogs.com/gif.latex?E_...7D]q_%7By+1%7D where http://www.codecogs.com/gif.latex?E_%7By%7D[X_%7Bn%7D] denotes the expectation of http://www.codecogs.com/gif.latex?X_n for a Markov chain starting at y. However, I haven't been able to get very far with this. One thing that is confusing me is how to relate the expectation of a variable in a chain starting in a particular state with the expectation used in the problem statement (which doesn't seem to specify which state the markov chain started in). Do I have to account for all the possible states that the chain might be in?

Any hints would be greatly appreciated--I'm a little frustrated right now.