Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.

As an example consider a woman who starts a small contracting business. She 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. In this section we will consider specific kinds of sequences that will allow us to calculate depreciation, such as the truck’s value.

Terms of an Arithmetic Sequence

The values of the truck in the example 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.

Example: Finding Common Differences

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

1. [latex]\left\{1,2,4,8,16,…\right\}[/latex]
2. [latex]\left\{-3,1,5,9,13,…\right\}[/latex]

Show Solution

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

1. The sequence is not arithmetic because there is no common difference.
   [latex]\begin{align}&2-1=1 && 4-2=2 && 8-4=4 && 16-8=8 \end{align}[/latex]
2. The sequence is arithmetic because there is a common difference. The common difference is 4.
   [latex]\begin{align}&1-(-3)=4 && 5-1=4 && 9-5=4 && 13-9=4 \end{align}[/latex]

Analysis of the Solution

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

Writing Terms of Arithmetic Sequences

Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of [latex]n[/latex] and [latex]d[/latex] into formula below.

[latex]{a}_{n}={a}_{1}+\left(n - 1\right)d[/latex]

Example: Writing Terms of Arithmetic Sequences

Write the first five terms of the arithmetic sequence with [latex]{a}_{1}=17[/latex] and [latex]d=-3[/latex].

Show Solution

Adding [latex]-3[/latex] is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.

The first five terms are [latex]\left\{17,14,11,8,5\right\}[/latex]

Analysis of the Solution

As expected, the graph of the sequence consists of points on a line.

Formulas for Arithmetic Sequences

Using Explicit Formulas for Arithmetic Sequences

We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.

[latex]{a}_{n}={a}_{1}+d\left(n - 1\right)[/latex]

To find the y-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.

The common difference is [latex]-50[/latex] , so the sequence represents a linear function with a slope of [latex]-50[/latex] . To find the [latex]y[/latex] -intercept, we subtract [latex]-50[/latex] from [latex]200:200-\left(-50\right)=200+50=250[/latex] . You can also find the [latex]y[/latex] -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis.

Recall the slope-intercept form of a line is [latex]y=mx+b[/latex]. When dealing with sequences, we use [latex]{a}_{n}[/latex] in place of [latex]y[/latex] and [latex]n[/latex] in place of [latex]x[/latex]. If we know the slope and vertical intercept of the function, we can substitute them for [latex]m[/latex] and [latex]b[/latex] in the slope-intercept form of a line. Substituting [latex]-50[/latex] for the slope and [latex]250[/latex] for the vertical intercept, we get the following equation:

[latex]{a}_{n}=-50n+250[/latex]

We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is [latex]{a}_{n}=200 - 50\left(n - 1\right)[/latex] , which simplifies to [latex]{a}_{n}=-50n+250[/latex].

Some arithmetic sequences are defined in terms of the previous term using a recursive formula. The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.

[latex]\begin{align}&{a}_{n}={a}_{n - 1}+d && n\ge 2 \end{align}[/latex]

Example: Writing a Recursive Formula for an Arithmetic Sequence

Write a recursive formula for the arithmetic sequence.

[latex]\left\{-18,-7,4,15,26, \ldots \right\}[/latex]

Show Solution

The first term is given as [latex]-18[/latex] . The common difference can be found by subtracting the first term from the second term.

[latex]d=-7-\left(-18\right)=11[/latex]

Substitute the initial term and the common difference into the recursive formula for arithmetic sequences.

[latex]\begin{align}&{a}_{1}=-18 \\ &{a}_{n}={a}_{n - 1}+11,\text{ for }n\ge 2 \end{align}[/latex]

Analysis of the Solution

We see that the common difference is the slope of the line formed when we graph the terms of the sequence, as shown in Figure 3. The growth pattern of the sequence shows the constant difference of 11 units.

Find the Number of Terms in an Arithmetic Sequence

Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.

In the following video lesson, we present a recap of some of the concepts presented about arithmetic sequences up to this point.

Solving Application Problems with Arithmetic Sequences

In many application problems, it often makes sense to use an initial term of [latex]{a}_{0}[/latex] instead of [latex]{a}_{1}[/latex]. In these problems we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:
[latex]{a}_{n}={a}_{0}+dn[/latex]

Key Equations

Key Concepts

• An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
• The constant between two consecutive terms is called the common difference.
• The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term.
• The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly.
• A recursive formula for an arithmetic sequence with common difference [latex]d[/latex] is given by [latex]{a}_{n}={a}_{n - 1}+d,n\ge 2[/latex].
• As with any recursive formula, the initial term of the sequence must be given.
• An explicit formula for an arithmetic sequence with common difference [latex]d[/latex] is given by [latex]{a}_{n}={a}_{1}+d\left(n - 1\right)[/latex].
• An explicit formula can be used to find the number of terms in a sequence.
• In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}+dn[/latex].

Glossary

arithmetic sequence a sequence in which the difference between any two consecutive terms is a constant

common difference the difference between any two consecutive terms in an arithmetic sequence