0

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 $$R_t=C_o+[C_o*P]+[[C_o*P]*P]+[[[C_o*P]*P]*P]...+$$

AlbertoM
  • 1
  • Welcome to MSE. I don't understand what $(R_1)=(C_o)$ is supposed to mean. It seems to say that the amount of refined material is equal to the amount of raw material, but that obviously isn't so. Also, you end up saying that $R_t=C_o*\displaystyle\sum_{i=0}^n{P^i}$ but $P$ gives the percent of raw material returned, not refined material produced. Is $R_t$ supposed to be $R_n$? So far as I can tell, $t$ appears nowhere else. Please try to clarify your question. – saulspatz Apr 06 '20 at 17:47
  • When the factory first refine an amout of raw material,$C_o$ obtain $R_1=C_o$ of refined material, plus raw material $C_1=C_o*P_o$, so $R_1$ is the first amount of refined material recived. $R_t$ is the sum of the refined materials in each step... the total refinded material... – AlbertoM Apr 06 '20 at 17:54
  • If the amount of refined material is equal to the amount of raw material, where does the additional raw material come from? Sounds like magic. – saulspatz Apr 06 '20 at 17:58
  • The amount of first refined material is equal to the first raw material to be refined (first step) then from this first step, the proccess return $C_1$ ,($C_1=C_0P^1$) of raw material. In the second step the returned raw material is refined.... and continues until $C_oP^n$<1 – AlbertoM Apr 06 '20 at 18:08

0 Answers0