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# Recognizing Arithmetic Sequences

A sequence is a finite or infinite ordered list of numbers, also called terms. An example of a sequence is $1,\, 4, \, 7, \, 10, \, 13, \ldots$

The dots indicate that the sequence continues infinitely, following the same pattern. In this sequence, the difference between each term is $3.$ Any sequence where the difference between consecutive terms is constant, is an arithmetic sequence.
Concept

## Patterns

In mathematics, a pattern describes a repeated change. In the example below, matches have been placed together to create three figures.

It is possible to find a pattern? Each figure has one more triangle than the last. Therefore, the next figure should have $4$ triangles.

The number of matches increases by $2,$ and the pattern will continue for the next figures. The first figure has $3$ matches, the second $5,$ etc.

Using this pattern, the fourth figure should have $9$ matches, the fifth $11,$ the sixth $13,$ etc. It can be described as the following arithmetic sequence:

$3,5,7,9,11,13,\ldots$
Concept

## Arithmetic Sequence

For an arithmetic sequence, the difference between consecutive terms is constant. Meaning, the difference between the first and second term is the same as the difference between the second and the third term, and so forth. This difference is called the common difference and is usually denoted with $d.$ An example of an arithmetic sequence is the following.

Here, the common difference is $d=2.$
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Exercise

Is the following sequence arithmetic? If so, write the next three terms in the sequence. $45,\ 39,\ 33,\ 27,\ 21,\ \ldots$

Show Solution
Solution

To determine if the sequence is arithmetic, we have to show that the difference between consecutive terms is constant. The difference between the first and second term is $39-45 = \text{-}6.$ The difference is negative, meaning that the second term is $6$ less than the first. If this is an arithmetic sequence the next term should be $39-6=33,$ and it is! Furthermore, $27$ is $6$ less than $33,$ and $21$ is $6$ less than $27.$ Thus, the sequence is arithmetic. To find the next term we subtract $6$ from $21\text{:}$ $21-6=15.$ The next term is $15.$ The one after that is $15-6=9,$ and the one after that is $9-6=3.$ To summarize, the sequence is arithmetic, and the next three terms are $15,$ $9,$ and $3.$

Concept

## Arithmetic Sequences and Linear Functions

Consider the arithmetic sequence $1,\ 2.5,\ 4,\ 5.5,\ 7, \ \ldots$ Here, the common difference is $1.5,$ and the terms can be illustrated in a table, where $n$ represents the term number and $a_n$ represents the term.

Because their terms change by a constant amount, arithmetic sequences show a linear relationship. Here, the common difference $d=1.5$ can be considered the slope of the line. In fact, when plotting an arithmetic sequence in a coordinate plane, it resembles the graph of a linear function.

By making this comparison, arithmetic sequences can be considered functions. They have the same characteristics as linear functions, but where linear functions are continuous, both the domain and range for arithmetic sequences are discrete.
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Exercise

In a theater, there are $10$ rows of seats. The first row has $11$ seats, and for each subsequent row, the number of seats increases by $3.$ Write an arithmetic sequence to represent the number of seats in each row, then graph the sequence.

Show Solution
Solution

To begin, we can start with the first row. It is given that this row has $11$ seats. Since each row has $3$ more seats than the previous row, we know the second row has $14$ seats. $11+3=14.$ Similarly, the third row has $17$ seats, because $14+3=17.$ Continuing this pattern for the remaining rows, we can write the following sequence. $11,\ 14,\ 17,\ 20,\ 23,\ 26,\ 29,\ 32,\ 35,\ 38.$ To graph the sequence, we can let $x$ be the row number and $y$ be the number of seats in the row. Arranging the sequence in a table can help us see the points that need to be graphed.

Lastly, to graph the sequence, we can plot the points from the table. Notice that the points aren't connected. This is because the row number and the number of seats in each row both have to be whole numbers.

The domain of the sequence is $D=\{1,2,3,4,5,6,7,8,9,10\},$ and the range is $R=\{11,14,17,20,23,26,29,32,35,38\}.$