General Rule for a sequence combining arithmetic and geometric progressions

Question

I am wanting to ask for a general rule to find the $n$th term of a sequence described by $$t_n = r (t_{n-1}) + c$$

Example: $120$ hectares of trees were in a forest. In an experiment, logging and replanting of trees took place. From Monday to Friday every week (starting from the first week), $3\%$ of the trees are logged, and during the weekends, $2.5$ hectares of trees would be planted. So if $W$ represents the number of remaining trees at the end of the $n$th week, the equation would be:

$W_n = 0.97 W_{n-1} + 2.5$, $W_0 = 120$

I am wanting to ask for the general rule for this sequence.

Thank you!

Answer 1

The solution will be the sum of two terms.

1. The general solution of the homogeneous equation $t_{n+1}=r\,t_n$. It will be of the form $C\,a^n$ for a certain constant $a$.
2. A particular solution of the complex equation $t_{n+1}=r\,t_n+c$. Since the non homogeneous term is a constant, you should look for a solution which is also a constant. This will work unless $r=1$, in which case the particular solution will be of the form $C\,n$ for a constant $C$.

If you have studied differential equations, you will recognize that the procedure is similar to the solution of linear equations with constant coefficients.