Solution:

The characteristic equation is,

$ \begin{array}{r} \\ (\lambda-3)=0 \\\\ \lambda=3 \\ \end{array} $

Hence the homogeneous solution is,

$ a_{r}^{(h)}=A(3)^{r} \\ $

Particular solution is of the type P (constant),

$ a_{r}=P \\ $

$ a_{r-1}=P \\ $

Substituting the value of $a_{r}$ and $a_{r-1}$ in the given recurrence relation.

$ \begin{aligned} P-3 P &=2 \\\\ -2 P &=2 \\\\ P &=-1 \\\\ a_{r} &=A(3)^{r}-1 \\ \end{aligned} $

Using initial condition,

$ \mathrm{a}_{0}=1 \\ $

$ \begin{aligned} 1 &=A-1 \Rightarrow A=2 \\\\ a_{r} &=2\left(3^{r}\right)-1 \\ \end{aligned} $