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 10.1.75a

Chapter 10, Problem 10.1.75a

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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Identify the initial amount of the drug in Jack's bloodstream, which is given as \(d_0 = 200\) mg at time \(n=0\) hours.

Understand that every hour, 5% of the drug is eliminated, meaning 95% remains. This implies the sequence is geometric with a common ratio \(r = 0.95\).

Write the general formula for the amount of drug after \(n\) hours as \(d_n = d_0 \times r^n = 200 \times (0.95)^n\).

Calculate the first five terms by substituting \(n = 0, 1, 2, 3, 4\) into the formula: \(d_0, d_1, d_2, d_3, d_4\).

Express each term as \(d_n = 200 \times (0.95)^n\) without simplifying the numerical values, to clearly show the pattern of the sequence.

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Key Concepts

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

Geometric Sequences

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 drug decreases by a fixed percentage each hour, making it a geometric sequence with a common ratio less than 1.

Exponential Decay

Exponential decay describes processes where quantities decrease at a rate proportional to their current value. Here, the drug amount decreases by 5% every hour, modeling exponential decay, which is key to understanding how the drug concentration changes over time.

Sequence Notation and Term Calculation

Sequence notation uses subscripts to denote terms, such as dₙ for the nth term. Calculating terms involves applying the decay factor repeatedly starting from the initial value, allowing us to write out specific terms like d₁, d₂, etc.

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Related Practice

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

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

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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.

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.

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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.

a. Find the first four terms of the sequence of heights {hₙ}.

h₀ = 20,r = 0.5

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Textbook Question

Explain why or why not

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

a.The sequence of partial sums for the series1 + 2 + 3 + ⋯ is {1, 3, 6, 10, …}.

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