## Arithmetic Sequences

### Learning Outcomes

By the end of this section, you will be able to:

• Find the common difference for an arithmetic sequence.
• Give terms of an arithmetic sequence.
• Write the formula for an arithmetic sequence.

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.

### A General Note: 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:

$\left\{{a}_{n}\right\}=\left\{{a}_{1},{a}_{1}+d,{a}_{1}+2d,{a}_{1}+3d,…\right\}$

### Example 1: Finding Common Differences

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

1. $\left\{1,2,4,8,16,…\right\}$
2. $\left\{-3,1,5,9,13,…\right\}$

### 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

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

$\left\{18,\text{ }16,\text{ }14,\text{ }12,\text{ }10,\dots \right\}$

### Try It

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

$\left\{1,\text{ }3,\text{ }6,\text{ }10,\text{ }15,\dots \right\}$

## 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 $n$ and $d$ into formula below.

${a}_{n}={a}_{1}+\left(n - 1\right)d$

### How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms.

1. Add the common difference to the first term to find the second term.
2. Add the common difference to the second term to find the third term.
3. Continue until all of the desired terms are identified.
4. Write the terms separated by commas within brackets.

### Example 2: Writing Terms of Arithmetic Sequences

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

### Try It

List the first five terms of the arithmetic sequence with ${a}_{1}=1$ and $d=5$ .

### How To: Given any the first term and any other term in an arithmetic sequence, find a given term.

1. Substitute the values given for ${a}_{1},{a}_{n},n$ into the formula ${a}_{n}={a}_{1}+\left(n - 1\right)d$ to solve for $d$.
2. Find a given term by substituting the appropriate values for ${a}_{1},n$, and $d$ into the formula ${a}_{n}={a}_{1}+\left(n - 1\right)d$.

### Example 3: Writing Terms of Arithmetic Sequences

Given ${a}_{1}=8$ and ${a}_{4}=14$ , find ${a}_{5}$.

### Try It

Given ${a}_{3}=7$ and ${a}_{5}=17$ , find ${a}_{2}$ .

## Using Formulas for Arithmetic Sequences

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.

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

### A General Note: Recursive Formula for an Arithmetic Sequence

The recursive formula for an arithmetic sequence with common difference $d$ is:

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

### How To: Given an arithmetic sequence, write its recursive formula.

1. Subtract any term from the subsequent term to find the common difference.
2. State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.

### Example 4: Writing a Recursive Formula for an Arithmetic Sequence

Write a recursive formula for the arithmetic sequence.

$\left\{-18,-7,4,15\,26,\dots\right\}$

### How To: Do we have to subtract the first term from the second term to find the common difference?

No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common difference.

### Try It

Write a recursive formula for the arithmetic sequence.

$\left\{25\text{, } 37\text{, } 49\text{, } 61\text{, } \dots\right\}$

## 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.

${a}_{n}={a}_{1}+d\left(n - 1\right)$

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 $-50$ , so the sequence represents a linear function with a slope of $-50$ . To find the $y$ -intercept, we subtract $-50$ from $200:200-\left(-50\right)=200+50=250$ . You can also find the $y$ -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in Figure 4.

Figure 4

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

${a}_{n}=-50n+250$

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 ${a}_{n}=200 - 50\left(n - 1\right)$ , which simplifies to ${a}_{n}=-50n+250$.

### A General Note: Explicit Formula for an Arithmetic Sequence

An explicit formula for the $n\text{th}$ term of an arithmetic sequence is given by

${a}_{n}={a}_{1}+d\left(n - 1\right)$

### How To: Given the first several terms for an arithmetic sequence, write an explicit formula.

1. Find the common difference, ${a}_{2}-{a}_{1}$.
2. Substitute the common difference and the first term into ${a}_{n}={a}_{1}+d\left(n - 1\right)$.

### Example 5: Writing the nth Term Explicit Formula for an Arithmetic Sequence

Write an explicit formula for the arithmetic sequence.

$\left\{2\text{, }12\text{, }22\text{, }32\text{, }42\text{, }\dots \right\}$

### Try It

Write an explicit formula for the following arithmetic sequence.

$\left\{50,47,44,41,\dots \right\}$

## Finding the Number of Terms in a Finite 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.

### How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.

1. Find the common difference $d$.
2. Substitute the common difference and the first term into ${a}_{n}={a}_{1}+d\left(n - 1\right)$.
3. Substitute the last term for ${a}_{n}$ and solve for $n$.

### Example 6: Finding the Number of Terms in a Finite Arithmetic Sequence

Find the number of terms in the finite arithmetic sequence.

$\left\{8\text{, }1\text{, }-6\text{, }\dots\text{, }-41\right\}$

### Try It

Find the number of terms in the finite arithmetic sequence.

$\left\{6,11,16,\dots,56\right\}$

## Solving Application Problems with Arithmetic Sequences

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

${a}_{n}={a}_{0}+dn$

### Example 7: Solving Application Problems with Arithmetic Sequences

A five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of$2 per week.

1. Write a formula for the child’s weekly allowance in a given year.
2. What will the child’s allowance be when he is 16 years old?

### Try It

A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?

## Key Equations

 recursive formula for nth term of an arithmetic sequence ${a}_{n}={a}_{n - 1}+d\phantom{\rule{1}{0ex}}n\ge 2$ explicit formula for nth term of an arithmetic sequence $\begin{array}{l}{a}_{n}={a}_{1}+d\left(n - 1\right)\end{array}$

## 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 $d$ is given by ${a}_{n}={a}_{n - 1}+d,n\ge 2$.
• As with any recursive formula, the initial term of the sequence must be given.
• An explicit formula for an arithmetic sequence with common difference $d$ is given by ${a}_{n}={a}_{1}+d\left(n - 1\right)$.
• 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 ${a}_{n}={a}_{0}+dn$.

## 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