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Sequences

Add or multiply the same number to get the next term in a sequence.

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Arithmetic and Geometric Sequences

A sequence is a list of numbers with a common pattern.  The common pattern in an arithmetic sequence is that the same number is added or subtracted to each number to produce the next number.  The common pattern in a geometric sequence is that the same number is multiplied or divided to each number to produce the next number.

Are all sequences arithmetic or geometric?

Sequences

A sequence is just a list of numbers separated by commas.  A sequence can be finite or infinite.  If the sequence is infinite, the first few terms are followed by an ellipsis \begin{align*}(\ldots)\end{align*} indicating that the pattern continues forever.

An infinite sequence: \begin{align*} 1, 2, 3, 4, 5, \ldots\end{align*}

A finite sequence: \begin{align*}2, 4, 6, 8\end{align*}

In general, you describe a sequence with subscripts that are used to index the terms.  The \begin{align*}k^{th}\end{align*} term in the sequence is \begin{align*}a_k\end{align*}.

\begin{align*}a_1, a_2, a_3, a_4,\ldots, a_k, \ldots\end{align*}

Arithmetic Sequences

Arithmetic sequences are defined by an initial value \begin{align*}a_1\end{align*} and a common difference \begin{align*}d\end{align*}.

\begin{align*}a_1 &=a_1 \\ a_2 &=a_1+d \\ a_3 &=a_1+2d \\ a_4 &=a_1+3d \\ & \ \ \vdots \\ a_n &=a_1+(n-1)d\end{align*}

    

Geometric Sequences

Geometric sequences are defined by an initial value \begin{align*}a_1\end{align*} and a common ratio \begin{align*}r\end{align*}.

\begin{align*}a_1 &=a_1\\ a_2 &=a_1 \cdot r\\ a_3 &=a_1\cdot r^2\\ a_4 &=a_1 \cdot r^3\\ & \ \ \vdots \\ a_n &=a_1 \cdot r^{n-1}\end{align*}

     

When trying to determine what kind of sequence it is, first test for a common difference and then test for a common ratio. If the sequence does not have a common difference or ratio, it is neither an arithmetic or geometric sequence.

\begin{align*}0.135, 0.189, 0.243, 0.297, \ldots\end{align*}  is an arithmetic sequence because the common difference is 0.054.

\begin{align*}\frac{2}{9}, \frac{1}{6}, \frac{1}{8}, \ldots\end{align*} is a geometric sequence because the common ratio is \begin{align*}\frac{3}{4}\end{align*}.

\begin{align*}0.54, 1.08, 3.24, \ldots\end{align*} is not arithmetic because the differences between consecutive terms are 0.54 and 2.16 which are not common.  The sequence is not geometric because the ratios between consecutive terms are 2 and 3 which are not common.

Examples

Example 1

Earlier, you were asked if all sequences are arithmetic or geometric. The sequence above shows that not all sequences are arithmetic or geometric.  Two famous sequences that are neither arithmetic nor geometric are the Fibonacci sequence and the sequence of prime numbers.

Fibonacci Sequence: \begin{align*} 1, 1, 2, 3, 5, 8, 13, 21, 34, \ldots\end{align*}

Prime Numbers: \begin{align*}2, 3, 5, 7, 11, 13, 17, 19, 23, \ldots\end{align*}

Example 2

For the following sequence, determine the common ratio or difference and the next three terms.

\begin{align*}\frac{2}{3}, \frac{4}{9}, \frac{6}{27}, \frac{8}{81}, \frac{10}{243}, \ldots\end{align*}

This sequence is neither arithmetic nor geometric.  The differences between the first few terms are \begin{align*}-\frac{2}{9}, -\frac{2}{9}, -\frac{10}{81}, -\frac{14}{243}\end{align*}.  While there was a common difference at first, this difference did not hold through the sequence.  Always check the sequence in multiple places to make sure that the common difference holds up throughout.

The sequence is also not geometric because the ratios between the first few terms are \begin{align*}\frac{2}{3}, \frac{1}{2}, \frac{4}{9}\end{align*}.  These ratios are not common.

Even though you cannot get a common ratio or a common difference, it is still possible to produce the next three terms in the sequence by noticing the numerator is an arithmetic sequence with starting term of 2 and a common difference of 2.  The denominators are a geometric sequence with an initial term of 3 and a common ratio of 3.  The next three terms are:

\begin{align*}\frac{12}{3^6}, \frac{14}{3^7}, \frac{16}{3^8}\end{align*}

Example 3

What is the tenth term in the following sequence?

\begin{align*}-12, 6, -3, \frac{3}{2}, \ldots\end{align*}

The sequence is geometric and the common ratio is \begin{align*}-\frac{1}{2}\end{align*}.  The equation is:

\begin{align*} a_n=-12 \cdot \left(-\frac{1}{2}\right)^{n-1}\end{align*}

The tenth term is:

\begin{align*}-12 \cdot \left(-\frac{1}{2}\right)^9=\frac{3}{128}\end{align*}

Example 4

What is the tenth term in the following sequence?

\begin{align*}-1, \frac{2}{3}, \frac{7}{3}, 4, \frac{17}{3}, \ldots\end{align*}

The pattern might not be immediately recognizable, but try ignoring the \begin{align*}\frac{1}{3}\end{align*} in each number to see the pattern a different way.

\begin{align*}-3, 2, 7, 12, 17, \ldots\end{align*}

You should see the common difference of 5.  This means the common difference from the original sequence is \begin{align*}\frac{5}{3}\end{align*}.  The equation is \begin{align*}a_n=-1+(n-1)\left(\frac{5}{3}\right)\end{align*}.  The \begin{align*}10^{th}\end{align*} term is:

\begin{align*}-1+9 \cdot \left(\frac{5}{3}\right)=-1+3 \cdot 5=-1+15=14\end{align*}

Example 5

Find an equation that defines the \begin{align*}a_k\end{align*} term for the following sequence.

\begin{align*}0, 3, 8, 15, 24, 35, \ldots\end{align*}

The sequence is not arithmetic nor geometric.  It will help to find the pattern by examining the common differences and then the common differences of the common differences.  This numerical process is connected to ideas in calculus.

0, 3, 8, 15, 24, 35

3, 5, 7, 9, 11

2, 2, 2, 2

Notice when you examine the common difference of the common differences the pattern becomes increasingly clear.  Since it took two layers to find a constant function, this pattern is quadratic and very similar to the perfect squares.

\begin{align*}1, 4, 9, 16, 25, 36, \ldots\end{align*}

The \begin{align*}a_k\end{align*} term can be described as \begin{align*}a_k=k^2-1\end{align*}

Review

Use the sequence \begin{align*}1, 5, 9, 13, \ldots\end{align*} for questions 1-3.

1. Find the next three terms in the sequence.

2. Find an equation that defines the \begin{align*}a_k\end{align*} term of the sequence.

3. Find the \begin{align*}150^{th}\end{align*} term of the sequence.

Use the sequence \begin{align*}12, 4, \frac{4}{3}, \frac{4}{9}, \ldots\end{align*} for questions 4-6.

4. Find the next three terms in the sequence.

5. Find an equation that defines the \begin{align*}a_k\end{align*} term of the sequence.

6. Find the \begin{align*}17^{th}\end{align*} term of the sequence.

Use the sequence \begin{align*}10, -2, \frac{2}{5}, -\frac{2}{25}, \ldots\end{align*} for questions 7-9.

7. Find the next three terms in the sequence.

8. Find an equation that defines the \begin{align*}a_k\end{align*} term of the sequence.

9. Find the \begin{align*}12^{th}\end{align*} term of the sequence.

Use the sequence \begin{align*}\frac{7}{2}, \frac{9}{2}, \frac{11}{2}, \frac{13}{2}, \ldots\end{align*} for questions 10-12.

10. Find the next three terms in the sequence.

11. Find an equation that defines the \begin{align*}a_k\end{align*} term of the sequence.

12. Find the \begin{align*}314^{th}\end{align*} term of the sequence.

13. Find an equation that defines the \begin{align*}a_k \end{align*} term for the sequence  \begin{align*}4, 11, 30, 67, \ldots\end{align*}

14. Explain the connections between arithmetic sequences and linear functions.

15. Explain the connections between geometric sequences and exponential functions.

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.2. 

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Vocabulary

arithmetic sequence

An arithmetic sequence has a common difference between each two consecutive terms. Arithmetic sequences are also known are arithmetic progressions.

common difference

Every arithmetic sequence has a common or constant difference between consecutive terms. For example: In the sequence 5, 8, 11, 14..., the common difference is "3".

common ratio

Every geometric sequence has a common ratio, or a constant ratio between consecutive terms. For example in the sequence 2, 6, 18, 54..., the common ratio is 3.

Explicit

Explicit formulas define each term in a sequence directly, allowing one to calculate any term in the sequence without knowing the value of the previous terms.

Explicit formula

Explicit formulas define each term in a sequence directly, allowing one to calculate any term in the sequence without knowing the value of the previous terms.

geometric sequence

A geometric sequence is a sequence with a constant ratio between successive terms. Geometric sequences are also known as geometric progressions.

sequence

A sequence is an ordered list of numbers or objects.

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