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Arithmetic Sequences

Sequences with a common difference

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Arithmetic Sequences

Do you know how to figure out a pattern? Take a look at this dilemma.


Use this Concept to figure out the following arithmetic sequence.


Look at this sequence.

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

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

\begin{align*} 2, 4, 6, 8, 10\end{align*}

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, right? Just like if you added another number to the sequence, you’d write 12. You noticed that there was a difference of 2 between each two numbers, or terms, in the sequence.

When we have a sequence with a fixed number between each of the terms, we call this sequence an arithmetic sequence.

Take a look at this one.

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

The difference is 5 between each number.

This is an arithmetic sequence. You can see that you have to be a bit of a detective to figure out the number patterns.

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

Take a look at this one.

Graph the sequence 2, 5, 8, 11, 14, 17,...

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

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—only the exact points are shown. 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 we have graphed. This is called continuous data and usually involves things like temperature or length that can change fractionally.

So, we 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.

Identify the pattern in the following sequences.

Example A

\begin{align*}3, 7, 11, 15\end{align*}

Solution: Add four

Example B

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

Solution: Add  -10

Example C

\begin{align*}81, 86, 91, 96\end{align*}

Solution: Add five

Now let's go back to the dilemma from the beginning of the Concept.


If we look for the difference between the values, you will see that each value has 4 added to it to equal the next value.

The pattern is add 4.

This is our answer.


Sequence - an ordered list of numbers, formed according to a rule
Arithmetic Sequence - a sequence where terms are formed by adding a common number each time
Common Difference (d) - the number that is added repeatedly to form an arithmetic sequence

Guided Practice

Here is one for you to try on your own.

What is the common difference in the following sequence?

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


You can see that positive two is added to the first value to find the second value. This happens for each of the values in the sequence.

Video Review

Arithmetic Sequences


Directions: 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

Directions: 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 \begin{align*} \frac{1}{4}\end{align*} of a caramel, the second day \begin{align*} \frac{1}{2}\end{align*} of a caramel, the third day \begin{align*} \frac{3}{4}\end{align*}.

  1. What is the difference between each day?
  2. How many do you think they’ll eat on the fourth, fifth, and sixth days?

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