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 10.2.71a

Chapter 10, Problem 10.2.71a

{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.

a.Find a recurrence relation for the sequence {dₙ} that gives the amount of drug in the blood after the nᵗʰ dose, where d₁ = 80.

Verified step by step guidance

Understand the problem context: The person takes 80 mg of aspirin every 24 hours, and the drug's half-life is 24 hours, meaning after 24 hours, half of the drug currently in the blood remains.

Define the sequence {dₙ} where dₙ represents the amount of aspirin in the blood immediately after the nᵗʰ dose is taken.

Recognize that between doses, the amount of drug in the blood reduces to half due to the half-life property. So, just before the nᵗʰ dose, the amount of drug is half of the amount after the (n-1)ᵗʰ dose, which is \( \frac{1}{2} d_{n-1} \).

When the nᵗʰ dose of 80 mg is taken, it adds to the remaining drug in the blood. Therefore, the amount after the nᵗʰ dose is \( d_n = \frac{1}{2} d_{n-1} + 80 \).

State the initial condition given: \( d_1 = 80 \), since after the first dose, the amount in the blood is exactly 80 mg.

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Key Concepts

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

Recurrence Relations

A recurrence relation defines each term of a sequence based on previous terms. In this problem, the amount of drug after each dose depends on the remaining drug from the previous dose plus the new dose. Understanding how to express this relationship mathematically is key to modeling the drug concentration over time.

Half-Life and Exponential Decay

Half-life is the time required for a substance to reduce to half its initial amount. Here, aspirin’s half-life of 24 hours means the drug amount halves every day. This concept helps determine how much drug remains in the blood before the next dose, forming the decay factor in the recurrence.

Sequences and Series in Modeling

Sequences represent ordered lists of numbers, often modeling quantities over time. In this context, the sequence {dₙ} tracks the drug amount after each dose. Recognizing how to use sequences to model repeated dosing and elimination is essential for predicting long-term drug levels.

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Related Practice

Textbook Question

18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.

a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.

∑ (k = 0 to ∞) (–2/7)ᵏ

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Textbook Question

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

a. Find an upper bound for the remainder in terms of n.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

154

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Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

a.Write out the first five terms of 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

71. Evaluating an infinite series two ways

Evaluate the series

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

a. Use a telescoping series argument.

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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.

a. Use Sₙ to estimate the sum of the series.

39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2