I have a problem and I am not able to find the solution, I hope someone can give me a little boost, it is a recurrence relationship, it says like this:
In a factory, material is refined, raw material enters and it is refined.
Every $C_o$ amount of raw material, the refinery returns $(R_1)=(C_o)$ of refined material plus $C_1=C_o*P_o$ of raw material.
$P$ is a number less than 1.
$C$ is the amount of raw material.
$R$ is the amount of refined material
The problem is: given a total amount of refined material $R_t$ to get and a fixed rate of return $P[$, find the amount of raw material $C_o$ necessary. Taking into account that they cannot return less than a 1 raw material.
I understand that the total of refined material follows the relationship: $$R_t=C_o*\displaystyle\sum_{i=0}^n{P^i}$$ Being $n$ such a number that $C_o*P^n<1$
But I don't know how to find $n$ to find $C_o$.
Thanks greetings.
EDIT: i see that $$R_t=C_o*\displaystyle\sum_{i=0}^n{P^i}$$ is wrong because if the process cannot return raw material less than 1 it must be $$R_t=C_o+[C_o*P]+[[C_o*P]*P]+[[[C_o*P]*P]*P]...+$$