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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 10.1.29c
Chapter 10, Problem 10.1.29c

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

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Identify the pattern in the given sequence: {1, 2, 4, 8, 16, ...}. Notice how each term relates to the previous one.
Observe that each term is obtained by multiplying the previous term by 2. This suggests the sequence is geometric with common ratio r = 2.
Recall the explicit formula for the nth term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio.
Substitute the known values into the formula: \(a_1 = 1\) and \(r = 2\), so the formula becomes \(a_n = 1 \times 2^{n-1}\).
Simplify the formula to get the explicit nth term: \(a_n = 2^{n-1}\). This formula allows you to find any term in the sequence directly.

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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 progress helps in identifying patterns and formulating explicit expressions.
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Explicit Formula for a Sequence

An explicit formula expresses the nth term of a sequence directly in terms of n, allowing calculation of any term without knowing previous terms. For example, geometric sequences have formulas like aₙ = a₁ * r^(n-1), where r is the common ratio.
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Geometric Sequences

A geometric sequence is one where each term is found by multiplying the previous term by a constant ratio. Recognizing this pattern is key to writing the explicit formula. In the given sequence {1, 2, 4, 8, 16, ...}, the ratio is 2, indicating a geometric progression.
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Related Practice
Textbook Question

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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Textbook Question

67–70. Formulas for sequences of partial sums Consider the following infinite series.


c.Make a conjecture for the value of the series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

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.



c.Find the limit of the sequence. What is the physical meaning of this limit?

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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.


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

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


c. A series that converges conditionally must converge.

Textbook Question

87. Explain why or why not

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


c. If ∑ aₖ converges, then ∑ (aₖ + 0.0001) converges.