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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.1.75a
Capitolo 10, Problema 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.

Guida verificata passo dopo passo
1
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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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.
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Geometric Sequences - Recursive Formula

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.
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Exponential Growth & Decay

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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Introduction to Sequences
Pratica correlata
Domanda del libro di testo

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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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

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Explain why or why not

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

a. If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series.

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


a. A series that converges must converge absolutely.

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