Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.2.71c

Chapter 10, Problem 10.2.71c

{Use of Tech} Periodic dosing
Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated.

c.Assuming the sequence has a limit, confirm the result of part (b) by finding the limit of {dₙ} directly.

Verified step by step guidance

Recognize that the amount of aspirin in the blood right after each dose forms a sequence {dₙ}, where each term represents the total amount of aspirin immediately after taking the nth dose.

Express the recursive relationship for the sequence: after 24 hours, half of the aspirin remains, and then 80 mg is added. This can be written as \(d_{n} = \frac{1}{2} d_{n-1} + 80\).

To find the limit of the sequence {dₙ}, denote the limit by \(L\). Since the sequence converges, the limit satisfies the same recursive formula: \(L = \frac{1}{2} L + 80\).

Solve the equation for \(L\) by isolating \(L\): subtract \(\frac{1}{2} L\) from both sides to get \(L - \frac{1}{2} L = 80\), which simplifies to \(\frac{1}{2} L = 80\).

Multiply both sides by 2 to find \(L = 160\). This confirms the limit of the sequence, representing the steady-state amount of aspirin in the blood after many doses.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Geometric Sequences and Series

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant ratio. In this problem, the amount of aspirin remaining after each 24-hour period forms a geometric sequence due to the half-life decay. Understanding how to express and sum geometric sequences is essential to analyze the drug concentration over time.

Limits of Sequences

The limit of a sequence describes the value that the terms approach as the number of terms goes to infinity. For the aspirin dosing problem, finding the limit of the sequence representing drug concentration helps determine the steady-state amount in the bloodstream after many doses. This concept is key to confirming long-term behavior of the drug level.

Half-Life and Exponential Decay

Half-life is the time required for a substance to reduce to half its initial amount, modeling exponential decay. Here, aspirin’s half-life of 24 hours means the drug amount halves every day. This decay rate defines the common ratio in the geometric sequence and is crucial for setting up the problem mathematically.

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Related Practice

Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

c.Find a recurrence relation that generates the sequence.

Drug elimination

Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

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Textbook Question

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

c.The convergent sequences {aₙ} and {bₙ} differ in their first 100 terms, but aₙ = bₙ for n > 100.

It follows that limₙ→∞aₙ = limₙ→∞bₙ.

Textbook Question

39–40. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, use Theorem 10.13 to complete the following.

c. Find lower and upper bounds (Lₙ and Uₙ, respectively) for the exact value of the series.

39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

c. Find an explicit formula for the nth term of the sequence.

{1, 3, 9, 27, 81, ......}

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Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

c. Find an explicit formula for the nth term of the sequence.

{-5, 5, -5, 5, ......}

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Textbook Question

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.

41. ∑ (k = 1 to ∞) 1 / k⁶