A system is written
$$ S_n = f^nS_0 \\ P_n = p(f-1)f^{n-1}S_0 $$
for $n\leq T$. For $n\geq T$:
$$ S_{n+1} = f(S_n-P_n ) \\ P_{n+1} = p(f-1)(S_n-P_n ) $$
The constants $S_0,f,p$ are all positive real (and $p<1$). I would like to obtain a general equation for $S_n$ and $P_n$ at time $n\geq T$. (No need to worry about $n\leq T$ since the equations are already in terms of the constants alone.) Is this possible?
Summation, product etc operators are fine
Hint: What can you say about $\frac{S_n}{P_n}$?