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 10.1.75c

Chapter 10, Problem 10.1.75c

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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Identify the initial condition given: the amount of drug at time zero is \(d_0 = 200\) mg.

Understand the process: every hour, 5% of the drug is eliminated, meaning 95% remains after each hour.

Express the amount of drug remaining after one hour in terms of the previous amount: \(d_1 = 0.95 \times d_0\).

Generalize this relationship to a recurrence relation for any hour \(n\): \(d_n = 0.95 \times d_{n-1}\) for \(n \geq 1\).

Summarize the recurrence relation with the initial condition: \(d_0 = 200\) and \(d_n = 0.95 \times d_{n-1}\).

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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 using one or more previous terms. It provides a way to model processes that evolve step-by-step, such as drug concentration over time. Understanding how to express the current amount based on the previous amount is key to formulating the relation.

Exponential Decay

Exponential decay describes a quantity that decreases by a fixed percentage over equal time intervals. In this problem, 5% of the drug is eliminated each hour, meaning 95% remains. Recognizing this helps to model the drug amount as a sequence that decreases multiplicatively.

Initial Conditions in Sequences

Initial conditions specify the starting value of a sequence, which is essential for solving recurrence relations. Here, d₀ = 200 mg sets the initial drug amount in the bloodstream, anchoring the sequence and allowing calculation of subsequent terms.

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Textbook Question

{Use of Tech} A savings plan

James begins a savings plan in which he deposits \(100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.

To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits \)100.

Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \(0.

c.How many months are needed to reach a balance of \)5000?

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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. If lim (as k → ∞) ᵏ√|aₖ| = 1/4, then ∑ 10aₖ converges absolutely.

Textbook Question

{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.

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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.If the terms of the sequence {aₙ} are positive and increasing, then the sequence of partial sums for the series∑⁽∞⁾ₖ₌₁aₖ diverges.

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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