Textbook Question
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.1.21
21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 2aₙ; a₁ = 2
Verified step by step guidance1
Identify the given recurrence relation: \(a_{n+1} = 2a_n\) with the initial term \(a_1 = 2\).
Use the initial term to find the second term by substituting \(n=1\) into the recurrence relation: \(a_2 = 2a_1\).
Find the third term by substituting \(n=2\): \(a_3 = 2a_2\).
Find the fourth term by substituting \(n=3\): \(a_4 = 2a_3\).
Write out the first four terms as \(a_1\), \(a_2\), \(a_3\), and \(a_4\) using the values found in the previous steps.
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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 one or more previous terms. It provides a way to generate the sequence step-by-step, starting from given initial conditions.
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Initial Conditions
Initial conditions specify the starting values of a sequence, which are necessary to compute subsequent terms using the recurrence relation. Without these, the sequence cannot be uniquely determined.
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Sequence Generation
Sequence generation involves applying the recurrence relation repeatedly to find terms beyond the initial ones. For example, using aₙ₊₁ = 2aₙ and a₁ = 2, we calculate a₂, a₃, and a₄ by substitution.
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Related Practice
Textbook Question
32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏ / √(k³ᐟ² + k)
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)k! / (kᵏ + 3)
Textbook Question
48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
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Textbook Question
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 20 / (∛k + √k)
Textbook Question
32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
1 / 1! + 4 / 2! + 9 / 3! + 16 / 4! + ⋯