Textbook Question
Recursively Defined Sequences
In Exercises 101–108, assume that each sequence converges and find its limit.
a₁ = 5,aₙ₊₁ = √(5aₙ)
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.1.22
Finding a Sequence’s Formula
In Exercises 13–30, find a formula for the nth term of the sequence.
2, 6, 10, 14, 18, …Every other even positive integer
Verified step by step guidance1
Identify the pattern in the sequence: 2, 6, 10, 14, 18, ... Notice that each term increases by a constant difference of 4, which means this is an arithmetic sequence.
Recall the general formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference.
Substitute the known values into the formula: the first term \(a_1 = 2\) and the common difference \(d = 4\), so the formula becomes \(a_n = 2 + (n - 1) \times 4\).
Simplify the expression by distributing the 4: \(a_n = 2 + 4n - 4\).
Combine like terms to write the formula in simplest form: \(a_n = 4n - 2\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arithmetic Sequences
An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant difference. Recognizing this pattern helps in formulating the nth term by identifying the first term and the common difference.
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Arithmetic Sequences - General Formula
General Formula for the nth Term
The nth term of an arithmetic sequence can be expressed as a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference. This formula allows direct calculation of any term in the sequence.
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Sequence Pattern Recognition
Identifying the pattern in the given sequence, such as recognizing it lists every other even positive integer, is essential. This insight guides the choice of the first term and common difference in the formula.
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Related Practice
Textbook Question
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ (n!)² / (2ⁿ (2n)!) ] xⁿ
Textbook Question
Finding nth Partial Sums
In Exercises 1–6, find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.
2 + (2/3) + (2/9) + (2/27) + … + (2 / 3ⁿ⁻¹) + …
Textbook Question
Using the Root Test
In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.
∑(from n=1 to ∞) [4ⁿ / (3n)ⁿ]
Textbook Question
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ / (1 + √n)]
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Textbook Question
Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.
6. (1 - x/3)^4