Problem 6 - Calculus: Early Transcendentals

Question

List the first five terms of the sequence. $$ a_n = \cos {n \pi}{2} $$

Step-by-Step Solution

Answer

The first five terms are 0, -1, 0, 1, 0.

Step 1: Understand the Sequence Formula

In the given sequence, the nth term is defined as \( a_n = \cos\left( \frac{n\pi}{2} \right) \). To find the first five terms, we will calculate the value of this expression for \( n = 1, 2, 3, 4, 5 \).

Step 2: Calculate the First Term

Substitute \( n = 1 \) into the formula: \( a_1 = \cos\left( \frac{1\pi}{2} \right) = \cos\left( \frac{\pi}{2} \right) = 0 \). Therefore, \( a_1 = 0 \).

Step 3: Calculate the Second Term

Substitute \( n = 2 \) into the formula: \( a_2 = \cos\left( \frac{2\pi}{2} \right) = \cos(\pi) = -1 \). Therefore, \( a_2 = -1 \).

Step 4: Calculate the Third Term

Substitute \( n = 3 \) into the formula: \( a_3 = \cos\left( \frac{3\pi}{2} \right) = \cos\left( \frac{3\pi}{2} \right) = 0 \). Therefore, \( a_3 = 0 \).

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \) into the formula: \( a_4 = \cos\left( \frac{4\pi}{2} \right) = \cos(2\pi) = 1 \). Therefore, \( a_4 = 1 \).

Step 6: Calculate the Fifth Term

Substitute \( n = 5 \) into the formula: \( a_5 = \cos\left( \frac{5\pi}{2} \right) = \cos\left( \frac{5\pi}{2} \right) = 0 \). Therefore, \( a_5 = 0 \).

Key Concepts

Sequence Formula

A sequence formula is a mathematical expression that defines the terms of a sequence based on their position. In this exercise, the sequence is given by:\[ a_n = \cos\left( \frac{n\pi}{2} \right) \]This formula indicates that each term in the sequence, denoted by \(a_n\), is determined by plugging in different values of \(n\).
To generate the sequence:

• Start by assigning values to \(n\) starting from 1.
• Substitute each value into the formula to get the corresponding term.
• Calculate the resulting value using the cosine function.

Following these steps, you can find each term in the sequence systematically.

Trigonometric Functions

Trigonometric functions are a fundamental part of mathematics, especially when dealing with angles and periodic phenomena. These functions include sine, cosine, and tangent.
The cosine function, denoted as \( \cos \), outputs the cosine of a given angle, typically measured in radians.
Trigonometric functions are periodic;

• The cosine function repeats every \(2\pi\) radians.
• Understanding this periodicity helps predict outputs for specific values.
• These functions are essential in analyzing cyclic patterns and waves.

Using trigonometric functions, you can solve a variety of problems, including those involving sequences like in this exercise.

Cosine Function

The cosine function is fundamental in trigonometry, characterized by its wave-like curve on a graph. It is defined on the set of all real numbers, taking values between -1 and 1.
Key properties of the cosine function include:

• It is an even function, meaning \( \cos(-x) = \cos(x) \).
• The function completes one full cycle in an interval of \(2\pi\) radians.
• Specific angles \(\left( \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \right)\) produce outputs such as 0, -1, and 1.

In our sequence, the varying angle \( \frac{n\pi}{2} \) utilizes these special angles, resulting in a patterned sequence of terms like 0, -1, and 1. Understanding the unique values at these angles helps in predicting the behavior of the sequence over increasing \(n\).