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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 10.1.33c
Chapter 10, Problem 10.1.33c

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
c. Find an explicit formula for the nth term of the sequence.


{-5, 5, -5, 5, ......}

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Identify the pattern in the given sequence: {-5, 5, -5, 5, ...}. Notice that the terms alternate between -5 and 5.
Recognize that this is an alternating sequence where the sign changes every term, but the absolute value remains constant at 5.
Recall that an alternating sign can be represented using powers of -1, specifically \((-1)^n\) or \((-1)^{n+1}\), which alternate between -1 and 1 as n increases.
Choose the appropriate power of -1 to match the first term: since the first term is -5 when n=1, test \((-1)^n\) and \((-1)^{n+1}\) to see which one gives -1 at n=1.
Write the explicit formula for the nth term as \(a_n = 5 \times (-1)^n\) or \(a_n = 5 \times (-1)^{n+1}\), depending on which matches the initial term, ensuring the sign alternates correctly.

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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 pattern or rule. Each number in the sequence is called a term, denoted as aₙ, where n indicates the term's position. Understanding how terms change helps in identifying the sequence's behavior and formulating its general term.
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Explicit Formula for a Sequence

An explicit formula expresses the nth term of a sequence directly in terms of n, without needing previous terms. It allows calculation of any term independently. Finding this formula involves recognizing patterns such as arithmetic, geometric, or alternating signs in the sequence.
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Alternating Sequences and Sign Patterns

Alternating sequences switch signs between terms, often modeled using powers of -1. For example, (-1)ⁿ or (-1)ⁿ⁺¹ generates alternating positive and negative values. Identifying this pattern is key to writing an explicit formula for sequences like {-5, 5, -5, 5, ...}.
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Related Practice
Textbook Question

{Use of Tech} A savings plan

James begins a savings plan in which he deposits \(100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.

To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits \)100.


Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \(0.


c.How many months are needed to reach a balance of \)5000?

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Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


c.Find a recurrence relation that generates 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.

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Textbook Question

Explain why or why not

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


c.The convergent sequences {aₙ} and {bₙ} differ in their first 100 terms, but aₙ = bₙ for n > 100.

It follows that limₙ→∞aₙ = limₙ→∞bₙ.

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.


c.Assuming the sequence has a limit, confirm the result of part (b) by finding the limit of {dₙ} directly.

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Textbook Question

Explain why or why not

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


c.If the terms of the sequence {aₙ} are positive and increasing, then the sequence of partial sums for the series∑⁽∞⁾ₖ₌₁aₖ diverges.

Textbook Question

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

c. Find an explicit formula for the nth term of the sequence.


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

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