Markov Model to compute the probaility on the $n^{th}$ day
Yes, you just add up the probabilities of all the possibilities. However, there's a conceptually clearer way of doing it that's less error-prone.

The transition matrix tells you the probability of moving from one state to another in one step.  The probability that it rains the day after tomorrow given that it's nice today is, as you've seen, given by
$$\Pr(N\to N)\,\Pr(N\to R) + \Pr(N\to R)\,\Pr(R\to R) + \Pr(N\to S)\,\Pr(S\to R)\,.$$
If you look at this closely, you'll see that it's exactly the (Nice, Rain) entry of the square of the transition matrix.  Similarly, the $n$th power of the transition matrix tells you the probability of moving between two states in $n$&nbsp;steps.

Powers of the transition matrix can be computed efficiently with repeated squaring&nbsp;&ndash; for example $M^7 = (M^2)^2\times M^2\times M$ and there's the bonus that you get all the $n$-step probabilities instead of just one.