I have three functions with values given as $$\begin{align*}    P(0) &= 0 \quad &    P(i+1) &= 5M(i)\\    M(0) &= 1  \quad &    M(i+1) &= R(i) + 2P(i)\\    R(0) &= 3  \quad &    R(i+1) &= R(i) + 3P(i)\,.\end{align*}$$If it was a linear recurrence with one function I could solve it using matrices. But here I cannot bring it to a single relationship. Each one is interrelated with the others. How do I approach this problem?
How do I solve interdependent recurrence relations?
proof techniques;recurrence relation
Eliminate one recurrence at a time, by plugging in.For instance, you can use the equation $P(i)=5 M(i-1)$ to eliminate every instance of $P(\cdot)$ in each of the other recurrences.  Plugging in, we get$$M(i) = R(i-1) + 10 M(i-2)$$$$R(i) = R(i-1) + 15 M(i-2).$$Now you have two inter-related recurrence relations, instead of three; you've eliminated one.Do it one more time, say to remove each instance of $M(\cdot)$.  You get something like$$R(i) = R(i-1) + 15 R(i-3) + 150 M(i-4).$$Do it again, to remove the $M(\cdot)$:$$R(i) = R(i-1) + 15 R(i-3) + 150 R(i-5) + 1500 M(i-7).$$Keep doing this, and you'll get a summation, say something like:$$R(i) = R(i-1) + 15 R(i-3) + 150 R(i-5) + \cdots + 15 \times 10^{(i-3)/2} \times M(0)$$(if $i$ is odd.)Now you're down to a single recurrence relation.  Solve it.  Then, use the equations above to derive the solutions for $P(\cdot)$ and $M(\cdot)$.