Recurrence Problem involving multiple dependencies. I have 3 equations :-


*

*$r_n=r_{n-1}+5m_{n-1}$

*$m_n = r_{n-1} + 3m_{n-1}$

*$p_n = 5m_{n-1}$


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.

It is given that,
$$r_n=r_{n-1}+5m_{n-1}$$ 
$$m_n = r_{n-1} + 3m_{n-1}.$$ 
By subtracting we can obtain that,
$$r_n=m_n+2m_{n-1}$$
$$r_{n-1}=m_{n-1}+2m_{n-2}.$$
Hence
$$m_n=4m_{n-1}+2m_{n-2}.$$
This second order liner recurrence with $m_0=1$ and $m_1=6.$
I think you can solve this one and try to find $r_n$ and $p_n$ via substituting $m_n.$
