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# Arithmetic Sequences

## Sequences with a common difference

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Arithmetic Sequences

The following picture represents a pattern. The pattern is an arithmetic sequence. Do you know how to figure out a pattern? Can you graph the pattern?

In this concept, you will learn to recognize, extend and graph arithmetic sequences.

### Arithmetic Sequence

Consider the following images.

You probably saw a pattern right away. If there were another set of boxes, you could probably guess at how many there would be.

If you saw this same pattern in terms of number of rectangles, it would look like this:

2, 4, 6, 8, 10

This set of numbers is called a sequence; it is a series of numbers that follow a pattern.

If there was another set of boxes, you’d probably guess there would be 12. If you added another number to the sequence, you would write 12. There was a difference of 2 between each two numbers, or terms, in the sequence.

When you have a sequence with a fixed number between each of the terms (a common difference), you call this sequence an arithmetic sequence.

Let’s look at an example.

What is the common difference between each of the terms in the following sequence?

7, 12, 17, 22, 27

First, subtract term 1 from term 2.

127=5\begin{align*}12-7 = 5\end{align*}

Next, subtract term 2 from term 3.

1712=5\begin{align*}17-12=5\end{align*}

Then, continue this subtraction for the rest of the sequence. The table below shows the difference between successive terms in the sequence.

 Terms Difference 7 12 12−7=5\begin{align*}12-7 = 5\end{align*} 17 17−12=5\begin{align*}17-12=5\end{align*} 22 22−17=5\begin{align*}22-17 = 5\end{align*} 27 27−22=5\begin{align*}27-22=5\end{align*}

The common difference for this sequence is 5.

This is an arithmetic sequence.

Finding the difference between two terms in a sequence is one way to look at sequences. You have used tables of values for several types of equations and you have used those tables of values to create graphs. Graphs are helpful because they are visual representations of the same numbers. When values rise, you can see them rise on a graph. Let’s use the same ideas, then, to graph arithmetic sequences. Let’s look at an example.

Graph the sequence 2,5,8,11,14,17,\begin{align*}2, 5, 8, 11, 14, 17, \ldots\end{align*}

First convert it into a table of values with independent values being the term number and the dependent values being the actual term.

Next, use this table to create a graph.

You can see the pattern clearly in the graph. That is one of the wonderful things about graphing arithmetic sequences.

In the graph that we created in the example, each term was expressed as a single point. This is called discrete data. Graphs of discrete data have only the exact points shown. You do not connect the dots with a line. This type of data is usually involves things that are counted in whole numbers like people or boxes.

Depending on what type of situation you are graphing, you may choose to connect the points with a line. The line demonstrates that there are data points between the points that you have graphed. This is called continuous data and usually involves things like temperature or length that can change fractionally.

So, you can graph sequences and classify them as either discrete or continuous data. Yet another possibility is continuing a sequence in either direction by adding terms that follow the same pattern.

### Examples

#### Example 1

Earlier, you were given a problem about the dot pattern.

You need to determine the pattern for the image below and then graph it.

First, let’s draw a table to show the term number, the number of dots, and the common difference.

 Term # # of dots Difference 1 3 2 5 5−3=2\begin{align*}5-3=2\end{align*} 3 7 7−5=2\begin{align*}7-5=2\end{align*} 4 9 9−7=2\begin{align*}9-7=2\end{align*} 5 11 11−9=2\begin{align*}11-9=2\end{align*}

The common difference for this sequence is 2.

This is an arithmetic sequence where you add 2 to get the next term in the sequence.

Next, graph the term number versus the number of dots.

Remember that these are discrete points and you are counting dots.

#### Example 2

What is the common difference in the following sequence?

15,13,11,9\begin{align*}-15, -13, -11, -9 \ldots\end{align*}

Use a table to find the difference between successive terms.

 Terms Difference -15 -13 −13−(−15)=2\begin{align*}-13-(-15)= 2\end{align*} -11 −11−(−13)=2\begin{align*}-11-(-13)=2\end{align*} -9 −9−(−11)=2\begin{align*}-9-(-11)=2\end{align*}

The common difference for this sequence is 2.

This is an arithmetic sequence where you add 2 to get the next term in the sequence.

#### Example 3

What is the common difference in the following sequence?

3, 7, 11, 15

Use a table to find the difference between successive terms.

 Terms Difference 3 7 7−3=4\begin{align*}7-3=4\end{align*} 11 11−7=4\begin{align*}11-7=4\end{align*} 15 15−11=4\begin{align*}15-11=4\end{align*}

The common difference for this sequence is 4.

This is an arithmetic sequence where you add 4 to get the next term in the sequence.

#### Example 4

What is the common difference in the following sequence?

18,8,2\begin{align*}18, 8, -2\end{align*}

Use a table to find the difference between successive terms.

 Terms Difference 18 8 8−18=10\begin{align*}8-18=10\end{align*} -2 −2−8=−10\begin{align*}-2-8=-10\end{align*}

The common difference for this sequence is -10.

This is an arithmetic sequence where you subtract 10 to get the next term in the sequence.

#### Example 5

What is the common difference in the following sequence?

81, 86, 91, 96

Use a table to find the difference between successive terms.

 Terms Difference 81 86 86−81=5\begin{align*}86-81=5\end{align*} 91 91−86=5\begin{align*}91-86=5\end{align*} 96 96−91=5\begin{align*}96-91=5\end{align*}

The common difference for this sequence is 5.

This is an arithmetic sequence where you add 5 to get the next term in the sequence.

### Review

Write the common difference for each sequence. If there is not a pattern, indicate this in your answer.

1. -9, -7, -5, -3, -1

2. 5.05, 5.1, 5.15, 5.2, 5.25

3. 3, 6, 10, 15, 21, 28

4. 17, 14, 11, 8, 5, 2

5. 10, 9, 8, 7, 6

6. 3, 5, 7, 9, 11

7. 3, 9, 27

8. 4, 8, 16, 32

9. 2, 3, 5, 9

10. 5, 11, 23, 47

11. 16, 8, 4, 2

12. 5, 10, 15, 20

13. 3, 6, 9, 12

Solve this problem by using what you know about arithmetic sequences.

An ant colony invades the caramels in a candy store. The first day they eat a 14\begin{align*}\frac{1}{4}\end{align*} of a caramel, the second day 12\begin{align*}\frac{1}{2}\end{align*} of a caramel, the third day 34\begin{align*}\frac{3}{4}\end{align*}.

14. What is the difference between each day?

15. How many do you think they’ll eat on the fourth, fifth, and sixth days?

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### Vocabulary Language: English

TermDefinition
arithmetic sequence An arithmetic sequence has a common difference between each two consecutive terms. Arithmetic sequences are also known are arithmetic progressions.
sequence A sequence is an ordered list of numbers or objects.

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