Problem 72 - Precalculus: Mathematical 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) Approximately 99.9023 mg after 10 doses; (b) 100 mg as the long-term limit.

Step 1: Understanding the Series

The total amount of drug in the bloodstream after each dose is determined by a series. The formula given is \(\sum_{k=1}^{n} 50\left(\frac{1}{2}\right)^{k-1} \). This represents a geometric series with the first term \(a = 50\) and a common ratio \(r = \frac{1}{2}\).

Step 2: Finding the Sum for Specific \(n\)

To calculate the amount of drug in the bloodstream after \(n = 10\) doses, use the sum formula for a finite geometric series: \[ S_n = a \frac{1-r^n}{1-r} \]Substituting \(a = 50\), \(r = \frac{1}{2}\), and \(n = 10\):\[ S_{10} = 50 \frac{1-\left(\frac{1}{2}\right)^{10}}{1-\frac{1}{2}} \]

Step 3: Calculating for \(n=10\)

Let's compute the value for \(S_{10}\):\[ S_{10} = 50 \frac{1-\frac{1}{1024}}{\frac{1}{2}} = 50 \times 2 \left(1 - \frac{1}{1024}\right) \]\[ S_{10} = 100 \left(1 - \frac{1}{1024}\right) = 100 \times \frac{1023}{1024} \]\[ S_{10} \approx 99.9023 \text{ mg} \]

Step 4: Evaluating the Infinite Series

For a geometric series, if \(|r| < 1\), the infinite series sum can be calculated by:\[ S = \frac{a}{1-r} \]Substitute \(a = 50\) and \(r = \frac{1}{2}\):\[ S = \frac{50}{1-\frac{1}{2}} = \frac{50}{\frac{1}{2}} = 100 \]

Step 5: Final Calculation and Results

For \(n=10\), the amount of drug in the bloodstream is approximately 99.9023 mg. For long-term use, the amount of drug in the bloodstream approaches 100 mg.

Key Concepts

Finite Geometric Series

A finite geometric series is a sum that consists of terms in a geometric progression, where you only add up a specific number of terms. Each term in the sequence is obtained by multiplying the previous term by a constant known as the common ratio.
For example, in our drug concentration problem, the series involves doses over a set time frame, specifically up to the 10th dose. Here, the first term is 50 mg and the common ratio is 0.5.
To find the sum of a finite geometric series, we use the formula:

• Sum: \[ S_n = a \frac{1-r^n}{1-r} \]
• Where \(a\) is the first term,
  \(r\) is the common ratio,
  and \(n\) is the number of terms.

For our series, the calculation shows that after 10 doses, the sum is approximately 99.9023 mg. This calculation accounts for how each additional dose contributes slightly less to the total concentration due to the diminishing effect of the drug's ongoing presence.

Infinite Series

Infinite series, unlike the finite kind, don't stop at a set number of terms—they go on forever. When dealing with geometric series, if the common ratio is a number between -1 and 1, the series converges to a specific value.
This is what we see in the continuous administration of the drug over a prolonged period. Even though the drug is administered daily without end, the series of doses doesn't lead to an infinite concentration in the bloodstream. Instead, it stabilizes.
For an infinite geometric series, the sum is found using the formula:

• Sum: \[ S = \frac{a}{1-r} \]
• Where \(a\) is the initial term, and \(r\) is the common ratio.

In our case, the drug reaches a stable level around 100 mg. This happens because as more doses are added, each contributes less and less, making the total sum approach a limit.

Sum of a Series

The sum of a series can be calculated using different formulas, depending on whether the series is finite or infinite. These sums tell us the total impact or accumulation over time.
For a finite geometric series, like our initial doses up to day 10, we can precisely calculate the total amount:

• The formula: \[ S_n = a \frac{1-r^n}{1-r} \]

This formula accounts for the contribution of each dose up to the specified day.
Meanwhile, for the infinite series related to the long-term drug administration, an infinite sum provides the "steady-state" value:

• The formula: \[ S = \frac{a}{1-r} \]

These sums are critical in fields like pharmacology and finance, where understanding the cumulative effect of repeated actions or transactions is important. Approximating infinite sums helps in knowing the limit we approach with ongoing procedures or investments.