College Algebra, Chapter 9, 9.3, Section 9.3, Problem 76

A certain drug is administered once a day. The concentration of the drug in the patient's bloodstream increases rapidly at first, but each successive does has less effect than the preceeding one. The total amount of the drug (in mg) in the bloodstream after the $n$th dose is given by

$\displaystyle \sum_{k = 1}^n 50 \left( \frac{1}{2} \right)^{k-1}$

a.) Find the amount of the drug in the bloodstream after $n = 10$ days.

b.) If the drug is taken on a long-term basis, the amount in the bloodstream is approximated by the infinite series $\displaystyle \sum^{\infty}_{k = 1} 50 \left( \frac{1}{2} \right)^{k-1}$. Find the sum of this series.

a.) After $n = 10$ days, the amount of drug in the bloodstream is


$
\begin{equation}
\begin{aligned}

\sum^{10}_{k = 1} 50 \left(\frac{1}{2} \right)^{k-1} =& 50 \left( \frac{1}{2} \right)^{1-1} + 50 \left( \frac{1}{2} \right)^{2-1} + 50 \left( \frac{1}{2} \right)^{3-1} + 50 \left( \frac{1}{2} \right)^{4 - 1} + 50 \left( \frac{1}{2} \right)^{5-1}+ 50 \left( \frac{1}{2} \right)^{6-1} + 50 \left( \frac{1}{2} \right)^{7-1} + 50 \left( \frac{1}{2} \right)^{8-1} + 50 \left( \frac{1}{2} \right)^{9-1} + 50 \left( \frac{1}{2} \right)^{10-1}
\\
\\
=& 99.90 \text{ mg}

\end{aligned}
\end{equation}
$


b.) Recall that the formula for the infinite geometric series is

$\displaystyle S = \frac{a}{1 - r}$

Notice that $a = 50$ and $\displaystyle r = \frac{1}{2}$. So,

$\displaystyle S = \frac{50}{\displaystyle 1 - \frac{1}{2}} = \frac{50}{\displaystyle \frac{1}{2}} = 100$

Therefore, the total amount of drug in the bloodstream after infinite days is 100 mg.