I have 3 equations :-

1. $r\_n=r\_{n-1}+5m\_{n-1}$
2. $m\_n = r\_{n-1} + 3m\_{n-1}$
3. $p\_n = 5m\_{n-1}$

The initial values of the sequences are

$$r\_0=3, m\_0=1, p\_0=0$$

How can I get the formula to get the nth term of the series without the recurrence?

A ready-made will be useful.