Problem 80 - Precalculus Mathematics for Calculus

Question

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 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) After 10 days, the amount is approximately 99.9 mg. (b) The infinite series sums to 100 mg.

Step 1: Identify the Series Type

The series given is a geometric series where each term takes the form \( a \cdot r^{k-1} \) with \( a = 50 \) and \( r = \frac{1}{2} \). For a geometric series, the sum of the first \( n \) terms, \( S_n \), is calculated using the formula: \[ S_n = a \frac{1-r^n}{1-r} \].

Step 2: Calculate the Sum for n=10

Use the formula for the sum of a finite geometric series to find the amount of the drug in the bloodstream after 10 days. Substitute \( a = 50 \), \( r = \frac{1}{2} \), and \( n = 10 \): \[ S_{10} = 50 \frac{1-(\frac{1}{2})^{10}}{1-\frac{1}{2}} \]. Calculate this value step-by-step.

Step 3: Compute Powers and Simplify

First, calculate \( (\frac{1}{2})^{10} \):1. \((\frac{1}{2})^{10} = \frac{1}{1024} = 0.0009765625\)Substitute back to the formula:2. \( S_{10} = 50 \frac{1-0.0009765625}{0.5} \) 3. \( S_{10} = 50 \frac{0.9990234375}{0.5} \)4. \( S_{10} = 50 \times 1.998046875 \)5. \( S_{10} \approx 99.90234375 \text{ mg} \).

Step 4: Sum of the Infinite Series

For an infinite geometric series, the sum is given by the formula \( S = \frac{a}{1-r} \). Substitute the values \( a = 50 \) and \( r = \frac{1}{2} \): \[ S = \frac{50}{1-\frac{1}{2}} = \frac{50}{0.5} = 100 \text{ mg} \].

Step 5: Interpret the Results

After 10 days, the concentration in the bloodstream is approximately 99.9 mg. Over an infinite period without changes in dosage or metabolism, the amount will stabilize at approximately 100 mg.

Key Concepts

Finite Geometric Series

In understanding a finite geometric series, it's essential to grasp what makes this type of series unique. A geometric series consists of terms where each term follows a specific pattern, multiplying the previous term by a common ratio, denoted as \( r \). For a finite geometric series, this sequence stops after a certain number of terms, \( n \). Each term in such a series can be represented using the formula \( a \cdot r^{k-1} \), where \( a \) is the first term.

To find the sum of a finite geometric series, the formula used is \( S_n = a \frac{1-r^n}{1-r} \). This formula allows you to add up all the terms in the series up to the \( n \)th term efficiently. For example, in a problem where a drug's concentration is measured over 10 days, substituting the appropriate values into this formula gives the total amount of the drug by the end of the 10-day period. It captures how this accumulation occurs step by step, emphasizing smaller additions as each day progresses due to the decreasing effect of each dose.

Infinite Geometric Series

An infinite geometric series extends endlessly, continually adding terms to the series. Despite having an infinite number of terms, such series can sometimes add up to a finite sum under certain conditions. This occurs when the absolute value of the common ratio \( r \) is less than 1 (\( |r|<1 \)).

The sum of an infinite geometric series is cleanly captured by the formula \( S = \frac{a}{1-r} \). For a practical example, consider a scenario where a drug's concentration is continually topped up in a patient's bloodstream. Over a prolonged period, the drug levels reach a steady state, dictated by the parameters of an infinite geometric series. This concept helps explain why the drug amount stabilizes at a constant sum, regardless of ongoing daily administration.

Sum of Series

The concept of sum is central in both finite and infinite geometric series. For a geometric series, the idea of summing involves adding all terms from beginning to end. Finite series limit these additions to a set number of terms, while infinite series continue indefinitely.

Calculating the sums allows us to understand better how quantities accumulate over time. The formulas provided are powerful tools, enabling us to handle these sums without manual, term-by-term addition. Whether it's monitoring drug concentration in the body or figuring out savings growth, the sum of series reveals critical insights and forecasts future states.

Convergence of Series

Convergence is a vital concept when dealing with infinite series. When an infinite geometric series converges, it means that even as you indefinitely continue adding terms, the series approaches a specific, finite limit.

This happens under the condition that the common ratio \( r \) satisfies \( |r|<1 \). When a series converges, it assures us that the additions, although infinite, do not spiral out of control. Instead, they huddle closer to a particular value. This has practical applications in numerous fields, from physics to finance, indicating how certain systems stabilize over time. It gives us the confidence to predict ongoing behavior, such as drug concentration stabilization, using straightforward mathematical principles.