4.16.2 What Is an Arithmetic Sequence?

Extending Your Thinking

Arithmetic Sequences

Finding Common Differences

Companies often make large purchases, such as computers and vehicles, for business use.

• A woman who starts a small contracting business purchases a new truck for $25,000.
• After five years, she estimates that she will be able to sell the truck for $8,000. The loss in value of the truck will therefore be $17,000, which is $3,400 per year for five years.
• The truck will be worth $21,600 after the first year; $18,200 after two years; $14,800 after three years; $11,400 after four years; and $8,000 at the end of five years.

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 common 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:

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

Example

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

1. {1, 2, 4, 8, 16, . . .}
2. {3, 1, 5, 9, 13, . . .}

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

a. The sequence is not arithmetic because there is no common difference.

$2-1=1$     $4-2=2$     $8-4=4$     $16-8=8$

b. The sequence is arithmetic because there is a common difference. The common difference is 4.

$1-(-3)=4$     $5-1=4$     $9-5=4$     $13-9=4$