How can I solve this infinite series?

Question

I'm trying to make a game which is based on custom-made economic system. I tried to make the value of currency increased by someone earning it, as the currency is decreased as the one takes it. And this is an example of the situation:

1. Let there is currency sized $50\text{T}$.
2. Let $A$ earns 10T by playing a game. Then there is currenty of left $40\text{T}$, and $10\text{T}$ becomes not able to earn by another player, because $A$ already took it. At this time, let $A$ did an effort sized $10\text{e}$. Namely, the value of $\text{T}$ is $1\text{e/T}$.
3. At this time, the currency left becomes $40\text{T}$, making the value of currency increasing.
4. Let $B$ did the same effort (amount of $10\text{e}$). $A$ earned $20\text{%}$ of the total amount of currency by the same effort with $B$, $B$ must earn the same cost that is corresponding to his effort.
5. At this time, left currency is $40\text{T}$, and $20\text{%}$ of that is $8\text{T}$, meaning that $B$ will earn $8\text{T}$ by doing the effort. Namely, the value of $\text{T}$ is $1.25 \text{e/T}$
6. And again, the currency becomes more worthy as left currency are lesser than before $B$ did his effort.

What the problem in this situation, is that these situations result different consequent:

• Doing effort of $10\text{e}$
• Doing effort of $5\text{e}$ and doing it again.
• Doing effort of $3\text{e}$, doing it extra twice times, and doing effort of $1\text{e}$

What I want to make is balance these situations to result in same consequent.

I tried to calculate how much the effort-maker will earn by creating the algebraical model, but I cannot make it be worked in my game, like in a program. These are the tries:

• Let $T\text{[T]}$ to be amount of currency that is able to be earned.
• Let $g\text{[e]}$ to be the absolute value of an effort.
• Let $a\text{[e]}$ to be the value of total $T$'s.
• Let $u\text{[e/T]}$ to be the unit value of the currency.
• Let $c\text{[T]}$ to be the cost of an effort of sized $g$.

What are in the brackets are units. Then, (TL;DR) \begin{equation} c=\lim_{n\rightarrow \infty} \sum_{k=1}^n c_k \end{equation} Where \begin{align} c_n&={\Delta g \over u(\Delta g, n)} \\ \Delta g&={g \over n} \\ u(\Delta g, n)&={a + \Delta g \over T_n} \\ T_n&=T-\sum_{k=1}^n c_k \end{align}

I want to compute the general term of $c$ when $T,g,a$ is defined. How can I do‽

P.s. I'm Korean so the translation may not be perfect. If somewhere is not understandable, please let me know by comment.