I need to find a limit involving a discrete finite state markov chain with absorbing state.

It is given that:

q(x,c) = r(x) + \beta E_{c'\lvert c}[q(x,c')]

so my thought are:

q(x,c') = r(x) + \beta E_{c''\lvert c'}[q(x,c'')]

such that :

q(x,c) = r(x) + \beta r(x) + \beta^2 E_{c''\lvert c}[q(x,c'')]

and repeating infinitely to get limit \frac{r(x)}{1-\beta}

but I do not know how to prove this ...

my guess is also that since there is a single absorbing state then:

\lim_{n \rightarrow \infty} E_{c^n\lvert c}[q(x,c^n)] = q(x,a)

with a being the "absorbing" value if that makes sense.