# 8.1: Arithmetic Sequences

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

The following sequence of numbers has a pattern you are bound to recognize:

$$2, 4, 6, 8, 10, 12, 14, 16, 18, …$$

Likely, you would describe the sequence in words: the sequence of even numbers. Alternatively, can we describe the sequence mathematically? That is, can we describe the pattern of the sequence of even numbers using a formula? Absolutely! This section will explore arithmetic sequences, how to identify them, mathematically describe their terms, and the relationship between arithmetic sequences and linear functions. Let’s get started!

##### Definition: Sequence

A sequence is a list of numbers: $$a_1 , a_2, a_3, a_4 , … , a_n, …$$ A sequence can be a finite or infinite list. We call $$a_1$$ the first term, $$a_2$$ the second term, and $$a_n$$ the “general term” or the $$n^{\text{th}}$$ term. Sequences have a pattern. We describe the pattern in the general term $$a_n$$.

For the sequence of even numbers: $$2, 4, 6, 8, 10, …$$ the general term $$a_n = 2n$$.

The general term $$a_n$$ of a sequence is simply a function of $$n$$, indicated above as $$f(n)$$, where $$n$$ is a natural number (or whole number if $$n$$ starts with zero).

##### Example 8.1.1

In the sequence of even numbers, what is the $$20^{\text{th}}$$ term in the sequence?

Solution

The general term of the sequence of even numbers is $$a_n = 2n$$. Since $$n =$$ the term number, we are asked to find $$a_{20}$$.

$$\begin{array}&& a_{20} = 2(20) = 40 &\text{Plug in the term-number \(n=20$$ into the formula $$a_n=2n$$} \end{array}\)

Answer The $$20^{\text{th}}$$ term of the sequence of even numbers is the number $$40$$.

##### Definition: Arithmetic Sequence

If the sequence: $$a_1, a_2, a_3, a_4 , … , a_{n−1}, a_n, …$$ exhibits a pattern such that

$a_n − a_{n−1} = d$

For all $$n$$, then the real number $$d$$ is called the common difference, and the sequence is an arithmetic sequence.

##### Example 8.1.2

A sequence is given. If the sequence is an arithmetic sequence, give the common difference. If the sequence is not an arithmetic sequence, explain how it fails to be arithmetic.

1. $$25, 32, 39, 46, 53, 60, …$$
2. $$2, 4, 8, 16, 32, …$$
3. $$3^2 , 3^4, 3^6, 3^8, 3^{10}, …$$
4. $$0, 1, 0, 1, 0, 1, …$$

Solution

1. Is the sequence

$$25, 32, 39, 46, 53, 60, …$$

an arithmetic sequence?

$$\begin{array} &a_2 − a_1 &= 32 − 25 &= \textcolor{red}{7} \\ a_3 − a_2 &= 39 − 32 &= \textcolor{red}{7} \\a_4 − a_3 &= 46 − 39 &= \textcolor{red}{7} \\a_5 − a_4 &= 53 − 46 &= \textcolor{red}{7} \\a_6 − a_5 &= 60 − 53 &= \textcolor{red}{7} \end{array}$$

The sequence is arithmetic and the common difference is $$7$$.

1. Is the sequence

$$2, 4, 8, 16, 32, …$$

an arithmetic sequence?

$$\begin{array} &a_2 − a_1 &= 4 − 2 &= \textcolor{red}{2} \\ a_3 − a_2 &= 8 − 4 &= \textcolor{red}{4} \\a_4 − a_3 &= 16 − 8 &= \textcolor{red}{8} \\a_5 − a_4 &= 32 − 16 &= \textcolor{red}{16} \\ &&\textcolor{red}{2 \neq 4 \neq 8 \neq 16} \end{array}$$

The sequence is not arithmetic. $$a_n − a_{n-1}$$ does not yield a common difference.

1. Is the sequence

$$3^2 , 3^4, 3^6, 3^8, 3^{10}, …$$

an arithmetic sequence?

$$\begin{array} &3^4 − 3^2 &= 3^2 (3^2 − 1) &= 9 \cdot 8 &= \textcolor{red}{72} \\ 3^6 − 3^4 &= 3^4 (3^2 − 1) &= 81 \cdot 8 &= \textcolor{red}{648} \end{array}$$

Since $$a_3 − a_2 \neq a_2 − a_1$$, we conclude the sequence is not arithmetic.

1. Is the sequence

$$0, 1, 0, 1, 0, 1, …$$

an arithmetic sequence?

$$\begin{array} 1-0 &= \textcolor{red}{1} \\ 0-1 &= \textcolor{red}{-1} \end{array}$$

Since $$a_3 − a_2 \neq a_2 − a_1$$, the sequence is not arithmetic.

## Find the General Term of an Arithmetic Sequence

If a sequence is arithmetic, the general term $$a_n$$ is determined using the common difference, $$d$$, of the sequence. Functions of the form $$y = mx+b$$, known as linear functions, have a strong relationship to arithmetic sequences. The slope $$m$$ of a linear function is equivalent to the common difference $$d$$ of an arithmetic sequence. Let’s compare arithmetic sequences to linear functions to build $$a_n$$, the general term of an arithmetic sequence.

##### Example 8.1.3

Find the general term $$a_n$$ of each arithmetic sequence:

1. $$4, 7, 10, 13, 16, …$$
2. $$100, 80, 60, 40, 20, …$$

Solution

We will create a table of values for each sequence. The first column will be the term number, $$n$$, starting with $$n = 1$$. The second column will list the terms of the sequence. The common difference is shown on the side of the second column.

1. The sequence $$4, 7, 10, 13, 16, …$$ has the common difference $$d = 3$$. But it’s also the slope $$m$$ of the linear function $$f(x) = mx + b$$.

$m = \dfrac{\delta y}{\delta x} = \dfrac{a_n − a_{n-1}}{n − (n − 1))} = \dfrac{d}{1} = d$

The above table essentially mimics any linear function, $$f(x) = mx+b$$.

• Instead of $$x$$, sequences use $$n$$−values.
• Instead of $$m =$$ slope in linear functions, sequences use $$d =$$ common difference.
• Instead of $$b$$, a sequence notates the same value with $$a_0$$.

If $$a_1$$ denotes the first term of a sequence, then the general term of a sequence is:

$a_n = f(n) = d \cdot n + a_0$

To find the general term, $$a_n$$, we will need to find the value $$a_0$$. There are several ways to do this, but perhaps the simplest is to create an extra row where $$n = 0$$, then use the common difference to find $$a_0$$. The common difference pattern is maintained and $$a_0 + d = a_1$$.

Find the value $$a_0$$:

$$\begin{array} &&a_0 + 3 &= 4 \\&a_0 + 3 − 3 &= 4 − 3 \\&a_0 &= 1\end{array}$$

The general term of the sequence is:

$$a_n = 3n + 1$$

1. Use the same strategy for Example $$8.1.3$$a to solve Example $$8.1.3$$b. Create a table, find the common difference, $$d$$, and find the $$a_0$$ term of the sequence $$100, 80, 60, 40, 20, …$$

The common difference $$d = −20$$. Find the value $$a_0$$.

$$\begin{array} &&a_0 − 20 &= 100 \\ &a_0 − 20 + 20 &= 100 + 20 \\ &a_0 &= 120 \end{array}$$

The general term of the sequence is:

$$a_n = −20n + 120$$

## Try It! (Exercises)

For #1-5, the general term of a sequence is given.

• List the first $$5$$ terms of the sequence: $$a_1, a_2, a_3, a_4, a_5$$.
• Is the sequence arithmetic?
1. $$a_n = n^2$$
2. $$a_n = 4 − 5n$$
3. $$a_n = 2^n$$
4. $$a_n = \dfrac{1}{2}n$$
5. $$a_n = 0.3n + 1$$

For #6-9, a table of values is given. State the general term, $$a_n$$, of the arithmetic sequence.

1. $$n$$ $$a_n$$
$$1$$ $$9$$
$$2$$ $$15$$
$$3$$ $$21$$
$$4$$ $$25$$
1. $$n$$ $$a_n$$
$$1$$ $$42$$
$$2$$ $$38$$
$$3$$ $$34$$
$$4$$ $$30$$
1. $$n$$ $$a_n$$
$$1$$ $$7$$
$$2$$ $$7.25$$
$$3$$ $$7.5$$
$$4$$ $$7.75$$
1. $$n$$ $$a_n$$
$$1$$ $$65.4$$
$$2$$ $$52.2$$
$$3$$ $$39$$
$$4$$ $$25.8$$

For #10-15, find the general term of the arithmetic sequence. Assume the first term is $$a_1$$.

1. $$8, 15, 22, 29, …$$
2. $$110, 85, 60, 35, …$$
3. $$9, 7.4, 5.8, 4.2, …$$
4. $$\dfrac{17}{2} , 8, \dfrac{15}{2} , 7, …$$
5. $$−20, −8, 4, 16, 28, …$$
6. $$4 \dfrac{1}{2} , 5 \dfrac{1}{4} , 6, 6 \dfrac{3}{4} , …$$

For #16-20, an arithmetic sequence is described. Find the general term $$a_n$$.

1. The arithmetic sequence has common difference $$d=8$$. The first term $$a_1 = 28$$.
2. The arithmetic sequence has first term $$a_1 = 40$$ and second term $$a_2 = 36$$.
3. The arithmetic sequence has first term $$a_1 = 6$$ and third term $$a_3 = 24$$.
4. The arithmetic sequence has common difference $$d = −2$$ and third term $$a_3 = 15$$.
5. The arithmetic sequence has common difference $$d = 3.6$$ and fifth term $$a_5 = 10.2$$.
1. Explain how the formula for the general term given in this section: $$a_n = d \cdot n + a_0$$ is equivalent to the following formula: $$a_n = a_1 + d(n − 1)$$
2. Some sequences have a finite number of terms. Find the number of terms in the finite arithmetic sequence: $$3, 17, 31, … ,143$$
3. Some sequences have a finite number of terms. Find the number of terms in the finite arithmetic sequence: $$80, 69, 58, … , −52$$
4. Create a formula for finding the number of terms a finite arithmetic sequence when given the first and the last term of the sequence. Assume the first term is $$a_1$$ and the last term is $$a_k$$.

This page titled 8.1: Arithmetic Sequences is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jennifer Freidenreich.