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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 10.1.31b
Chapter 10, Problem 10.1.31b

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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Identify the pattern in the given sequence: {1, 3, 9, 27, 81, ...}. Notice how each term relates to the previous term.
Observe that each term is obtained by multiplying the previous term by 3. For example, 3 = 1 \(\times\) 3, 9 = 3 \(\times\) 3, and so on.
Express this relationship as a recurrence relation: \[a_{n} = 3 \times a_{n-1}\] where \[n \geq 2\].
Specify the initial condition, which is the first term of the sequence: \[a_1 = 1\].
Combine the recurrence relation and the initial condition to fully define the sequence: \[a_1 = 1, \quad a_n = 3a_{n-1} \text{ for } n \geq 2\].

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Key Concepts

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

Sequences and Terms

A sequence is an ordered list of numbers defined by a specific rule. Each number in the sequence is called a term, denoted as aₙ, where n indicates the term's position. Understanding how terms relate to their positions is essential for analyzing and generating sequences.
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Introduction to Sequences

Recurrence Relation

A recurrence relation expresses each term of a sequence as a function of one or more previous terms. It provides a way to generate the sequence step-by-step, starting from initial term(s). Identifying the recurrence relation helps in understanding the pattern and behavior of the sequence.
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Initial Conditions

Initial conditions specify the starting term(s) of a sequence, which are necessary to uniquely determine all subsequent terms using the recurrence relation. Without these values, the sequence cannot be fully generated or analyzed.
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Related Practice
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

{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.

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

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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. The sum ∑ (k = 3 to ∞) 1 / √(k − 2) is a p-series.

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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