I would like to understand the proposed solution of this example.
I can't understand the $E[N_k|N_{k-1}]$ expression in the second line. Shouldn't that be, with total probability:
$$E[N_k|N_{k-1}] = p\ (1 + N_{k-1}) + (1 - p)\ (1 + E[N_k])$$
Where is the p-multiplicant before $(N_{k - 1} + 1)$ and why can he (Roos, Introduction to the Probability Models, p. 113) alternate between random variable ($N_{k - 1}$) and expectation ($E[N_k]$) in the same expression? The derived expression is correct, by the way. I verified it with a simulation and a more straightforward algebraic approach.