Arithmetic Sequence (Progression): Definition and Examples

Look closely at this series:

what do you notice?

It is very clear that there is some relationship between each number that follows it or the one before it. This relationship is that the difference between every two consecutive numbers is 4. That is, if we know the value of the term in the current position, we can know the value of the term in the next or previous position only by adding the number 4 to the value of the term in the current position.

This fixed number is called common difference and we denote it by d.

The number d determines the next term by adding d to the current term, and so on. This is how arithmetic sequences are generated.

There are three different cases of the number d:

If d > 0 then the sequence is increasing, that is, each term is greater than the term before it.
If d < 0 then the sequence is decreasing, that is, each term is less than the term before it.
If d = 0, then the sequence is constant.

Note:

An arithmetic sequence (i.e arithmetic progression) is in which the difference between each term the preceding term is always constant.

Practical examples of arithmetic sequences

Example1:

Find the missing number in the following arithmetic sequence:

Solution:

First we must calculate d (common difference)

In these variances we got a fixed number. So , which means that the sequence is increasing.

Now, in order to find the missing number, it is enough to add 5 to the number before the number we are looking for which is 16.

So the arithmetic sequence is:

Example2:

Find the missing number in the following arithmetic sequence:

Solution:

First we must calculate d (common difference):

In these variances we got a fixed number. So , which means that the sequence is decreasing.

Now, in order to find the missing number, it is enough to add -4 to the number before the number we are looking for which is -7.

So the arithmetic sequence is:

Example3:

Find the missing number in the following arithmetic sequence:

Solution:

First we must calculate d (common difference)

In these variances we got a fixed number. So , which means that the sequence is increasing.

Now, in order to find the missing number, it is enough to subtract 6 from the number after the number we are looking for which is 8.

So the arithmetic sequence is:

Example4:

Find the missing numbers in the following series for which they are an arithmetic sequence:

Solution:

In this case we should look for a number that will give us a steady pace to start from 6 and get to 34.

The method of calculating d in this case:

Note:

In which and two different terms of an arithmetic sequence.
The number represents the number of missing numbers between and .

We have in the example:

Then,

So the arithmetic sequence is: