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Chapter 15: Problem 1

What is an arithmetic sequence? Give an example.

Short Answer

Expert verified
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant, called the common difference (denoted as \(d\)). The general form is \(a_n = a_1 + (n - 1)d\). An example of an arithmetic sequence is \[ 3, 7, 11, 15, 19, \dots \] with a common difference of 4.

Step by step solution

01

Define Arithmetic Sequence

An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is always constant. This constant difference is called the common difference and is usually denoted by \(d\).
02

General Form

The general form of an arithmetic sequence can be written as: \(a_n = a_1 + (n - 1)d\), where \(a_n\) is the \(n\)-th term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.
03

Example of Arithmetic Sequence

We will provide an example of an arithmetic sequence. Consider the sequence: \[ 3, 7, 11, 15, 19, \dots \]
04

Verify the Common Difference

To confirm that this is an arithmetic sequence, we need to check if the difference between consecutive terms is constant: - Difference between the second term (7) and the first term (3) is \(7 - 3 = 4\). - Difference between the third term (11) and the second term (7) is \(11 - 7 = 4\). - Difference between the fourth term (15) and the third term (11) is \(15 - 11 = 4\). - Difference between the fifth term (19) and the fourth term (15) is \(19 - 15 = 4\). Since the difference between consecutive terms is constant and equal to 4, this sequence is an arithmetic sequence with a common difference of 4.

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