Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.2.71b

Chapter 10, Problem 10.2.71b

{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.

b.Use a calculator to estimate this limit. In the long run, how much drug is in the person’s blood?

Verified step by step guidance

Identify the problem as a periodic dosing model where a fixed amount of aspirin (80 mg) is taken every 24 hours, and the drug decays by half every 24 hours due to its half-life.

Express the amount of aspirin in the blood right after each dose as a sequence \( A_n \), where \( A_n \) is the amount immediately after the \( n^{th} \) dose.

Set up the recursive relation for the sequence: after 24 hours, half of the previous amount remains, and then 80 mg is added. This gives \( A_{n} = \frac{1}{2} A_{n-1} + 80 \).

Recognize that this is a linear difference equation and find its long-term behavior by solving for the steady-state (limit) \( L \) where \( L = \frac{1}{2} L + 80 \).

Solve the equation for \( L \) to find the equilibrium amount of aspirin in the blood after many doses, which represents the long-run amount of drug present.

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Key Concepts

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

Exponential Decay and Half-Life

Half-life is the time required for a quantity to reduce to half its initial value, modeling exponential decay. In this problem, aspirin’s half-life of 24 hours means the drug amount halves every day, which can be expressed using exponential decay formulas to track the drug concentration over time.

Geometric Series and Limits

Repeated dosing with decay leads to a sum of decreasing amounts forming a geometric series. Understanding how to sum an infinite geometric series and find its limit is essential to determine the steady-state amount of drug in the blood after many doses.

Steady-State Concentration in Pharmacokinetics

Steady-state concentration occurs when the amount of drug administered equals the amount eliminated over a dosing interval. Calculating this equilibrium helps estimate the long-term drug level in the bloodstream, crucial for understanding the drug’s effectiveness and safety.

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Textbook Question

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

b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.

41. ∑ (k = 1 to ∞) 1 / k⁶

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Textbook Question

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

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

{1, 2, 4, 8, 16, ......}

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Textbook Question

71. Evaluating an infinite series two ways

Evaluate the series

∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.

b. Use a geometric series argument with Theorem 10.8.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. The sum ∑ (k = 3 to ∞) 1 / √(k − 2) is a p-series.

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Textbook Question

{Use of Tech} Fibonacci sequence

The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about A.D. 1200 as a model for the growth of rabbit populations.

It is given by the recurrence relation: fₙ₊₁ = fₙ + fₙ₋₁,for n = 1, 2, 3, … where f₀ = 1 and f₁ = 1. Each term of the sequence is the sum of its two predecessors.

b.Is the sequence bounded?

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Textbook Question

57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.

b. Find an explicit formula for the nth term of the sequence {hₙ}.

h₀ = 30,r = 0.25

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