Problem 72 - Algebra and Trigonometry

Question

Drug Concentration \(\quad\) A certain drug is administered once a day. The concentration of the drug in the patient's blood-stream increases rapidly at first, but each successive dose has less effect than the preceding one. The total amount of the drug (in mg) in the bloodstream after the \(n\) th dose is given by $$\sum_{k=1}^{n} 50\left(\frac{1}{2}\right)^{k-1}$$ (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 \(\sum_{k=1}^{\infty} 50\left(\frac{1}{2}\right)^{k-1} .\) Find the sum of this series.

Step-by-Step Solution

Answer

(a) 99.9025 mg after 10 days; (b) 100 mg on long-term basis.

Step 1: Understand the Series

The given series for the total amount of the drug is \( \sum_{k=1}^{n} 50 \left( \frac{1}{2} \right)^{k-1} \). This is a finite geometric series where the first term \( a \) is 50, and the common ratio \( r \) is \( \frac{1}{2} \).

Step 2: Evaluate Finite Geometric Series for n=10

For a geometric series \( \sum_{k=1}^{n} ar^{k-1} \), the sum \( S_n \) can be calculated using the formula: \( S_n = a \frac{1-r^n}{1-r} \). Substituting \( a = 50 \), \( r = \frac{1}{2} \), and \( n = 10 \), we calculate:\[S_{10} = 50 \frac{1-(\frac{1}{2})^{10}}{1-\frac{1}{2}} = 50 \times \frac{1-\frac{1}{1024}}{\frac{1}{2}} = 100 \left(1 - \frac{1}{1024}\right)\]\[S_{10} = 100 \times \frac{1023}{1024} = 99.9025\] mg.

Step 3: Evaluate Infinite Geometric Series

The infinite series \( \sum_{k=1}^{\infty} 50 \left( \frac{1}{2} \right)^{k-1} \) is also a geometric series with \( a = 50 \) and \( r = \frac{1}{2} \). The sum \( S_\infty \) of an infinite geometric series is given by \( S_\infty = \frac{a}{1-r} \). Substituting the values:\[S_\infty = \frac{50}{1-\frac{1}{2}} = \frac{50}{\frac{1}{2}} = 100\] mg.

Key Concepts

Finite Geometric Series

A finite geometric series is a series with a fixed number of terms, usually described by its first term and a common ratio, which is the factor by which one term is multiplied to get the next term. In mathematical terms, a finite geometric series with first term \( a \), common ratio \( r \), and \( n \) terms is given by:

• \( S_n = a \frac{1-r^n}{1-r} \)

This formula helps us find the sum of all terms in the series. For example, to find the total drug concentration in the bloodstream after ten doses, you would plug in the values \( a = 50 \), \( r = \frac{1}{2} \), and \( n = 10 \) into the formula.
By substituting these values, you reach:

• \( S_{10} = 50 \frac{1-(\frac{1}{2})^{10}}{1-\frac{1}{2}} \)

Calculating gives us the amount \( 99.9025 \) mg, showing how the finite sum evaluates as each dose has successively less effect.

Infinite Geometric Series

An infinite geometric series extends indefinitely, implying the series has an infinite number of terms. Such a series can still have a sum, given certain conditions. For a geometric series with first term \( a \) and common ratio \( r \), it converges (i.e., has a finite sum) only if the absolute value of the common ratio is less than 1 \( |r| < 1 \).
The formula for the sum \( S_\infty \) is quite elegant:

• \( S_\infty = \frac{a}{1-r} \)

For example, in the context of the drug, \( a = 50 \), \( r = \frac{1}{2} \), which satisfies the convergence condition. Plugging these values in, we discover:

• \( S_\infty = \frac{50}{1 - \frac{1}{2}} = 100 \)

This shows the amount of drug stabilizes at 100 mg over the long-term, providing a safe and effective treatment guideline.

Sum of a Series

The sum of a series can be finite or infinite, often computed using well-defined formulas. Understanding how to find this sum is critical in many practical applications, like understanding drug concentrations in a medical setting.

• A finite series involves adding up all terms from the first to a certain number \( n \).
• An infinite series brings in the concept of convergence, which can lead to a finite sum even if the series appears to go on forever.

In both cases, the sum calculation provides insights into the compounded effects over the entire series span. For the drug example, using the geometric series formulas allows for transparent and straightforward calculations, leading to a practical understanding of how the drug accumulates in the bloodstream over time.