Skip to main content
Mathematics LibreTexts

8.1: Arithmetic Sequences

  • Page ID
    83157
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

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

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \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 \(a_n = 2n\).

    clipboard_eddc57d00c36a2fbed49c43f696c7302b.png

    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 \(f(x) = mx + b\).

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

    clipboard_ecbb8d4cf8b9fa5856cc56124c921e2e0.png

    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 \(a_0 + d = a_1\).

    clipboard_e6eccaea315b268044364f34c0948747d.png

    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 \(100, 80, 60, 40, 20, …\)

    clipboard_ee68ed069e37b2760f8fc0f27b5a37f04.png

    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.

    • Was this article helpful?