Problem 72 - College Algebra

Question

What is an arithmetic sequence? Give an example with your explanation.

Step-by-Step Solution

Answer

In mathematics, an arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant, this difference is typically referred to as the 'common difference'. For example, in the sequence \(2, 4, 6, 8, 10\), the common difference is \(2\).

Step 1: Definition of Arithmetic Sequence

An arithmetic sequence, often also called arithmetic progression, is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is typically referred to as the 'common difference'.

Step 2: Characteristics of an Arithmetic Sequence

The key characteristics of an arithmetic sequence are: 1) There must be at least two numbers in the sequence. 2) The difference between any two consecutive terms must be the same. This is called the 'common difference'.

Step 3: Example of an Arithmetic Sequence

For instance, consider the sequence \(2, 4, 6, 8, 10\). Indeed, it is an arithmetic sequence. The common difference here is \(2\) - each term is \(2\) more than the previous one. As you can see, \(4 - 2 = 2\), \(6 - 4 = 2\), \(8 - 6 = 2\), and \(10 - 8 = 2\). Thus, the rule of this arithmetic sequence could be formulated as 'start at \(2\) and add \(2\) at each step'.