Problem 41
Question
A patient takes 150 \(\mathrm{mg}\) of a drug at the same time every day. Just before each tablet is taken, 5\(\%\) of the drug remains in the body. (a) What quantity of the drug is in the body after the third tablet? After the \(n\) th tablet? (b) What quantity of the drug remains in the body in the long run?
Step-by-Step Solution
Verified Answer
After the 3rd tablet: 157.875 mg; Long run: 3000 mg.
1Step 1: Understand the Problem
Each day, 5% of the previous day's drug quantity remains, which means 95% of it is eliminated. Then, an additional 150 mg is added due to the new tablet.
2Step 2: Calculate Drug Quantity After Each Tablet
Let \( A_n \) be the amount of the drug in the body before the \( n \)-th tablet. We know that \( A_{n+1} = 0.05 A_n + 150 \).
3Step 3: Calculate Drug Quantity after the 3rd Tablet
Start by finding \( A_1 = 150 \), then \( A_2 = 0.05 \times 150 + 150 \). Continue to find \( A_3 = 0.05 \times A_2 + 150 \).
4Step 4: Create a Recursive Formula for Any n
The recursive formula is \( A_n = 0.05^{n-1} \times 150 + 150 \sum_{k=0}^{n-2} 0.05^k \). Use the geometric series formula to simplify this.
5Step 5: Simplify Using Geometric Series
The geometric series sum is \( S_n = \frac{1 - 0.05^{n-1}}{1 - 0.05} \). Substitute this in to find \( A_n = 150(0.05^{n-1} + 20(1 - 0.05^{n-1})) \).
6Step 6: Determine Long-Term Drug Quantity
As \( n \to \infty \), \( 0.05^{n-1} \to 0 \). The long-run drug quantity is \( A_{\infty} = 150 \times 20 = 3000 \).
Key Concepts
Recursive FormulasGeometric SeriesLong-Term Behavior
Recursive Formulas
Recursive formulas are mathematical expressions used to define sequences in a step-wise manner. Think of them like a set of instructions that tell you how to get from one term to the next. In the context of drug elimination, we are interested in finding out how much of a drug remains in a patient’s body each day given its regular administration and elimination.
A recursive formula allows you to compute the current value based on the previous one. For example, in this context, if we let \(A_n\) be the amount of drug in the body before the \(n\)-th tablet is taken, the recursive formula is defined as:
Understanding recursive formulas is essential because they show us the ongoing process or a pattern, helping us to predict future values without recalculating everything from the start.
A recursive formula allows you to compute the current value based on the previous one. For example, in this context, if we let \(A_n\) be the amount of drug in the body before the \(n\)-th tablet is taken, the recursive formula is defined as:
- \(A_{n+1} = 0.05A_n + 150\)
Understanding recursive formulas is essential because they show us the ongoing process or a pattern, helping us to predict future values without recalculating everything from the start.
Geometric Series
A geometric series is a series of terms that each multiply by a constant factor to obtain the next. This concept is pivotal for understanding how the drug builds up in the body over time in a pattern where each term is a fixed proportion of the previous one.
Here, when constructing the recursive formula, we observed a sum of a geometric series while analyzing the buildup of the drug: each term is smaller than the last by a factor of the drug retention rate, 0.05.
For example, if you have a geometric series \( 0.05^0, 0.05^1, 0.05^2, \ldots\) representing the drug amounts retained over successive days, you can use the formula to find the sum:
Here, when constructing the recursive formula, we observed a sum of a geometric series while analyzing the buildup of the drug: each term is smaller than the last by a factor of the drug retention rate, 0.05.
For example, if you have a geometric series \( 0.05^0, 0.05^1, 0.05^2, \ldots\) representing the drug amounts retained over successive days, you can use the formula to find the sum:
- \( S_n = \frac{1 - 0.05^{n-1}}{1 - 0.05} \)
Long-Term Behavior
Long-term behavior in a mathematical model indicates what happens to the sequence or series as time moves towards infinity. For the drug elimination model, we're interested in knowing the drug quantity that stabilizes in the body after many doses.
Using the recursive formula and understanding the geometric series, we can analyze the long-term drug behavior. We find that as \( n \to \infty \), the term \( 0.05^{n-1} \to 0 \). This suggests that the new terms added into the sequence are minuscule and have negligible impact over time.
Thus, the stable, long-term amount of the drug in the body is the result of a steady buildup balanced by the regular elimination. With the recursive formula used and applying geometric series simplification, the long-term behavior reveals a drug amount stabilizing at:
Using the recursive formula and understanding the geometric series, we can analyze the long-term drug behavior. We find that as \( n \to \infty \), the term \( 0.05^{n-1} \to 0 \). This suggests that the new terms added into the sequence are minuscule and have negligible impact over time.
Thus, the stable, long-term amount of the drug in the body is the result of a steady buildup balanced by the regular elimination. With the recursive formula used and applying geometric series simplification, the long-term behavior reveals a drug amount stabilizing at:
- \( A_{\infty} = 150 \times 20 = 3000 \) mg
Other exercises in this chapter
Problem 40
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Use the power series for tan \(^{-1} x\) to prove the following expression for \(\pi\) as the sum of an infinite series: $$ \pi=2 \sqrt{3} \sum_{n=0}^{\infty} \
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If \(\Sigma a_{n}\) is a convergent series with positive terms, is it true that \(\sum \sin \left(a_{n}\right)\) is also convergent?
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\(41-42=\) Let \(\left\\{b_{n}\right\\}\) be a sequence of positive numbers that con- verges to \(\frac{1}{2} .\) Determine whether the given series is absolute
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