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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.2.89a
Chapter 10, Problem 10.2.89a

{Use of Tech} Drug Dosing
A patient takes 75 mg of a medication every 12 hours; 60% of the medication in the blood is eliminated every 12 hours.




a.Let dₙ equal the amount of medication (in mg) in the bloodstream after n doses, where d₁ = 75.
Find a recurrence relation for dₙ.

Verified step by step guidance
1
Understand the problem: The patient takes 75 mg of medication every 12 hours, and 60% of the medication is eliminated every 12 hours. We want to find a recurrence relation for the amount of medication in the bloodstream after n doses, denoted as \(d_n\), with \(d_1 = 75\) mg.
Identify what happens between doses: After each 12-hour period, 60% of the medication is eliminated, so 40% remains. This means the amount of medication just before taking the next dose is 40% of the previous amount, or \(0.4 \times d_{n-1}\).
Account for the new dose: At the time of the nth dose, the patient takes an additional 75 mg, which adds to the remaining medication from the previous dose.
Write the recurrence relation: Combining the remaining medication and the new dose, the amount after the nth dose is given by \[d_n = 0.4 \times d_{n-1} + 75\]
Confirm the initial condition: The first dose amount is given as \(d_1 = 75\), which fits the recurrence relation when \(n=1\).

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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 previous terms. In this problem, the amount of medication after each dose depends on the remaining amount from the previous dose plus the new dose. Understanding how to express this relationship mathematically is key to modeling the drug concentration over time.
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Exponential Decay

Exponential decay describes how a quantity decreases by a consistent percentage over equal time intervals. Here, 60% of the medication is eliminated every 12 hours, meaning 40% remains. This decay factor is used to calculate the remaining drug amount before the next dose is added.
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Initial Conditions in Sequences

Initial conditions specify the starting value of a sequence, which is essential for solving recurrence relations. Given d₁ = 75 mg, this sets the baseline amount of medication after the first dose, allowing subsequent terms to be computed accurately.
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Related Practice
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 = 30,r = 0.25

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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. 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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Textbook Question

87. Explain why or why not

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


a. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 10 to ∞) aₖ converges.

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Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

a. Find the next two terms of the sequence.


{1, 3, 9, 27, 81, ......}

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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. A series that converges must converge absolutely.

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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. n!n! = (2n)! for all positive integers n.

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