# What is an Arithmetic Sequence?

**Sequences typically have set patterns that enable us to predict what each term might be.**

A list of numbers ordered in a particular way is a sequence, and each individual number is referred to as a term.

Understanding arithmetic sequences, and how to identify them is a great way to develop critical thinking skills. Identifying how numbers relate to each other, and what commonalities they have in a sequence can carry over into better critical thinking in other intellectual endeavors.

**An arithmetic sequence is a sequence or progression of numbers where the difference between each number is the same (or a constant).**

For example:

In this series 5, 12, 19, 26… We can tell that this is an arithmetic sequences by subtracting each number from the one following it.

5-12= -7t

12-19- - 7

19-26= -7

The difference is the same each time, therefore it is a constant.

**In other words, to tell whether or not it is an arithmetic sequence, we need to be able to see what is happening between each number, and whatever happens between two needs to be the same thing that is happening between each consecutive number.**

Another way to write this arithmetic sequence is:

( a, a+d, a + 2d, a +3d,…..)

The letter ‘a’ represents the first term, and the letter ‘d’ represents the constant difference.

So, looking back at our sequence of numbers let’s apply this:

5 , 12 , 19, 26

Using this method, let’s plug in our numbers.

We have ‘5’ as our first term, and we know that the constant difference is 7.

5, 5+7, 5 + 2(7) , 5 + 3(7) = 5, 12, 19, 26

What if we wanted to find the 50th term of this sequence?

Let’s write an arithmetic sequence as a formula:

Xn = a + d(n-1)

The reason “n-1” is used is because d, the constant difference, is not applied to the very first term.

Let’s take a look at a new arithmetic sequence, and, using our new formula, calculate a certain term.

Look at this sequence:

9, 17, 25, 33, 41, 49, 57, 65, 73, 81

So, we are going to calculate the 5th term using our formula. We know that our first term is 9, so now we have to calculate the constant difference, which is 8.

So we are looking for our 4th term which is represented by Xsub4, a is our first term (which is 9), d is our constant difference (which is 8), and n represents the numbers sequence order (which in this case is 4).

So, let’s plug in our numbers.

Xsub4= 9 + 8(4-1) = 33

Typically, you wouldn’t use this formula to calculate a term that is already listed, but rather to predict and calculate a term farther along the progression of numbers.

So, let’s say we want to calculate the 1,698th term of this arithmetic sequence.

Xsub1,698= 9 + 8(1,698-1)= 13,585

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