Solve $a_0 = 1$, and $a_n = 3a_{n-1}$, for $n \in \Bbb Z_+$ using its characteristic equation

Question

Solve $a_0 = 1$, and $a_n = 3a_{n-1}$, for $n \in \Bbb Z_+$ using the characteristic equation method. This is my attempt:

Characteristic polynomial is: $r^2 – 3r = r(r – 3) $

With roots $r_1 = 0$ and $r_2 = 3$.

Now I don't know do I need:

$a_m = α⸱0^m + β⸱3^m$

$a_0 = 1 = α⸱0^0 + β⸱3^0$

or only:

$a_0 = 1 = α⸱0^0$

and I don't know how to finish it :(