Concentration of a drug after repeated injections After injection of  a dose $D$ of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as $D\exp^{-at}$ where $t$ represents time in hours and $a$ is a positive constant.  
a) If a dose $D$ is injected every $T$ hours, write an expresssion of the sum of the residual concentrations just before the $(n+1)$st injection. 
b) Determine the limiting pre-injection concentration. 
c) If the concentration of insulin must always remain at or above a critical value $C$, determine the minimal dosage $D$ in terms of $C,a$ and $T$. 
What I got so far:
a) 
The sum can be expressed as $D\exp^{-a(t-0T)} + D\exp^{-a(t-1T)} + D\exp^{-a(t-nT)} \dots$
So it is The sum from $0$ to $\infty$ of $D\exp^{-a(t-nt)}$
I'm not sure If I am right.
b ) I think I just need to find the limit of this expression as $t$ goes to $0$? so it's simply $D$.
c) I don't know what to do here, but I tried looking up boundaries so the function is bound above some value $C$, etc. but I'm really lost here.

(a) We calculate the first few terms. Let $R_n$ be the concentration just before the $(n+1)$-th injection. 
We have $R_1=De^{-aT}$. Thus $R_2=De^{-aT}+e^{-aT}R_1=De^{-aT}+De^{-2aT}$.
Similarly, $R_3=De^{-aT}+e^{-aT}R_2=De^{-aT}+De^{-2aT}+De^{-3aT}$.
The pattern is clear. We have
$$R_n=De^{-aT}+De^{-2aT}+\cdots+De^{-naT}.$$
This is an $n$-term  geometric series with first term $De^{-aT}$ and common ratio $e^{-aT}$. The sum is given by
$$R_n=De^{-aT} \frac{1-e^{-naT}}{1-e^{-aT}}.\tag{1}$$ 
(b) Let $n\to\infty$. The $e^{-naT}$ term goes to $0$.  Now from (1) we can write down the limiting pre-injection concentration. 
(c) This seems too easy. The smallest  concentration (after treatment has begun) is just before the second injection. So we want $De^{-aT}\ge C$. The minimal dose therefore satisfies $De^{-aT}=C$. Solve for $D$. 
