For a four state $(s_1, s_2, s_3, s_4)$ Markov chain, the transition probability matrix is given by:
$$P = \begin{bmatrix} 1-a & a & 0 & 0\\ 1-b & 0 & b & 0\\ 1-c & 0 & 0 & c\\ 1 & 0 & 0 & 0 \end{bmatrix}$$
I need to calculate $f_1^{(n)}$, the probability that a first return to state $s_1$ occurs at the $n$-th step.
My thoughts were the following: I calculate $P^n$ and $P^{n-1}$ and then say that the probability that its first return to state $s_1$ occurs at the $n$-th step is $P^n_{s_1, s_1} - P^{n-1}_{s_1, s_1}$?
However, the assignment asks me to calculate the aforementioned probability from the transition diagram. Any ideas on how I could do that?
Modify the chain so that you make the state $s_1$ absorptive. That is, consider the matrix
$$P_1 = \begin{bmatrix} 1 & 0 & 0 & 0\\ 1-b & 0 & b & 0\\ 1-c & 0 & 0 & c\\ 1 & 0 & 0 & 0 \end{bmatrix}$$
Now the probability of the original chain returning to the state $s_1$ for the first time at step $n$ is the same as this new chain being in state $s_1$ at step $n-1$ and not being in it at the step $n-2$, when you give it the initial distribution of what the distribution will be in the original chain after $1$ step, that is
$$\begin{bmatrix} 1 - a\\ a\\ 0\\ 0 \end{bmatrix}$$
In other words, you assume one step is already taken in the absorptive chain.