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 10.1.75b

Chapter 10, Problem 10.1.75b

72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence

b.Find an explicit formula for the 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.

Verified step by step guidance

Identify the type of sequence described in the problem. Since the drug amount decreases by a fixed percentage each hour, this is a geometric sequence.

Determine the initial term of the sequence, which is the amount of drug at time zero: \(d_0 = 200\) mg.

Find the common ratio \(r\) of the geometric sequence. Since 5% of the drug is eliminated each hour, 95% remains, so \(r = 1 - 0.05 = 0.95\).

Write the explicit formula for the \(n\)-th term of the sequence using the geometric sequence formula: \(d_n = d_0 \times r^n\).

Substitute the known values into the formula to get \(d_n = 200 \times (0.95)^n\), which expresses the amount of drug in the bloodstream \(n\) hours after the dose.

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

Explicit Formula for Sequences

The explicit formula expresses the nth term of a sequence directly in terms of n, without needing previous terms. For a geometric sequence, the formula is dₙ = d₀ × rⁿ, where d₀ is the initial term and r is the common ratio.

Exponential Decay

Exponential decay describes processes where quantities decrease by a consistent percentage over equal time intervals. Here, the drug amount reduces by 5% each hour, so the remaining amount follows an exponential decay model.

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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

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

b. (2n)! / (2n − 1)! = 2n

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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.If a sequence of positive numbers converges, then the sequence is decreasing.

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Textbook Question

67–70. Formulas for sequences of partial sums Consider the following infinite series.

a.Find the first four partial sums S₁, S₂, S₃, S₄ of the series.

∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

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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 = 20,r = 0.5

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Textbook Question

Find the first term a and the ratio r of each geometric series.

a. ∑ k = 0 to ∞(2/3) × (1/5)ᵏ

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