Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.1.19

Chapter 10, Problem 10.1.19

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = 1 + sin(πn / 2)

Verified step by step guidance

Identify the given explicit formula for the sequence: \(a_n = 1 + \sin\left(\frac{\pi n}{2}\right)\), where \(n\) is a positive integer starting from 1.

Recall that to find the first four terms of the sequence, you need to substitute \(n = 1, 2, 3, 4\) into the formula one by one.

Calculate each term by plugging in the values of \(n\): - For \(n=1\), compute \(a_1 = 1 + \sin\left(\frac{\pi \times 1}{2}\right)\). - For \(n=2\), compute \(a_2 = 1 + \sin\left(\frac{\pi \times 2}{2}\right)\). - For \(n=3\), compute \(a_3 = 1 + \sin\left(\frac{\pi \times 3}{2}\right)\). - For \(n=4\), compute \(a_4 = 1 + \sin\left(\frac{\pi \times 4}{2}\right)\).

Use your knowledge of sine values at special angles (multiples of \(\frac{\pi}{2}\)) to simplify each sine term without a calculator, for example, \(\sin\left(\frac{\pi}{2}\right)\), \(\sin(\pi)\), \(\sin\left(\frac{3\pi}{2}\right)\), and \(\sin(2\pi)\).

Write down the first four terms \(a_1, a_2, a_3, a_4\) explicitly after simplification to complete the sequence.

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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 or formula. Each number in the sequence is called a term, denoted as aₙ, where n indicates the position. Understanding how to find terms from the formula is essential to write out the sequence.

Explicit Formula for Sequences

An explicit formula directly defines the nth term of a sequence as a function of n, allowing calculation of any term without knowing previous terms. For example, aₙ = 1 + sin(πn/2) gives a direct way to find each term by substituting n.

Evaluating Trigonometric Functions at Specific Angles

To find terms involving sine functions, you must evaluate sin(θ) at specific angles, often multiples of π. Knowing values like sin(0) = 0, sin(π/2) = 1, sin(π) = 0, and sin(3π/2) = -1 helps compute terms accurately.

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Introduction to Trigonometric Functions

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