MA001: College Algebra

The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. Each term increases or decreases by the same constant value called the common difference of the sequence. For this sequence, the common difference is –3,400.

The sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any term of the sequence, and add 3 to find the subsequent term.

Arithmetic Sequence

An arithmetic sequence is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the common difference. If $a_1$ is the first term of an arithmetic sequence and $d$ is the common difference, the sequence will be:

$\{a_n\}=\{a_1,a_1+d,a_1+2d,a_1+3d,...\}$

Example 1

Finding Common Differences

Is each sequence arithmetic? If so, find the common difference.

Solution

Subtract each term from the subsequent term to determine whether a common difference exists.

Analysis

The graph of each of these sequences is shown in Figure 1. We can see from the graphs that, although both sequences show growth, $a$ is not linear whereas $b$ is linear. Arithmetic sequences have a constant rate of change so their graphs will always be points on a line.

Figure 1

Q&A

If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the common difference?

No. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract it from the subsequent term to find the common difference.

Try It #1

Is the given sequence arithmetic? If so, find the common difference.

${18,16,14,12,10,…}$

Try It #2

Is the given sequence arithmetic? If so, find the common difference.

$\{1,3,6,10,15,…\}$