For solving linear discrete time model of the form $$N_{t+1}=pN_{t}+c$$
(when saying solving I mean that $N_{t}$ is expressed only in function of $p$, $c$ and the initial conditions)
one can first solve for the equilibrium: $$N_{equi}=\frac{c}{1-p}$$
Then define a variable d as the distance of the system from the equilibrium:
$$D_{t}=N_{t}-N_{equi}$$
The recursion equation for D is $D_{t+1}=pD_{t}$ because $$D_{t+1}=n_{t+1}-N_{equi}=pN_{t}+c-N_{equi}$$
Replacing $N_{t}$ with $D_{t}-N_{equi}$ gives $$D_{t+1}=pD_{t}+pN_{equi}+c-N_{equi}$$
Therefore, the general solution for D is
$$D_{t}=p^tD_{0}$$
So, the general solution for Nis
$$N_{t}=p^t(N_{0}-N_{equi})+N_{equi}$$
which also equals
$$N_{t}=p^tc_{0}+(1-p^t)N_{equi}$$
Does it make sense?
What is the recipe for solving a model of the form: $$N_{t}=pN_{t-1}+qN_{t-T}+c$$
T can take any integer values in the range [0;t]
The answer is maybe given here by Ron Gordon but I could not really understand it.
$\displaystyle{N_{t} = p\,N_{t - 1} + q\,N_{t - T} + c}$
Define $\displaystyle{% \xi_{t} = N_{t} - {c \over 1 - p -q} \quad\Longrightarrow\quad \xi_{t} = p\,\xi_{t - 1} + q\,\xi_{t - T} }$. Try solutions like $\mu^{t}$ which leads to the characteristic equation $$ \mu^{T} - p\,\mu^{T - 1} - q = 0 $$
You get $T$ solutions: $\mu_{1},\mu_{2}\ldots\mu_{T}$. Your general solution looks like
$$ N_{t} = \sum_{n = 1}^{T}A_{n}\,\mu_{n}^{t} + {c \over 1 - p - q} $$