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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
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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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Exponential Growth & Decay

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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Introduction to Sequences
Related Practice
Textbook Question

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

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).


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

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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.


b.Use a calculator to estimate this limit. In the long run, how much drug is in the person’s blood?

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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 sum ∑ (k = 1 to ∞) 1 / 3ᵏ is a p-series.

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, 2, 4, 8, 16, ......}

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Textbook Question

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.


Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

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Textbook Question

{Use of Tech} Fibonacci sequence

The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about A.D. 1200 as a model for the growth of rabbit populations.


It is given by the recurrence relation: fₙ₊₁ = fₙ + fₙ₋₁,for n = 1, 2, 3, … where f₀ = 1 and f₁ = 1. Each term of the sequence is the sum of its two predecessors. 


b.Is the sequence bounded?

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