Arithmetic Mean

There is one more piece of terminology associated with arithmetic progressions that you have to learn - arithmetic mean. You have already studied the math behind this new concept (assuming that you have worked through the Arithmetic Terms Problems page).

Definition

An Arithmetic Mean is simply a term BETWEEN two other terms in an Arithmetic Progression.

Look at the following A.P.:

2, 6, 10, ...

In this progression, 6 is an arithmetic mean of 2 and 10. In another example:

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

3, 5, and 7 are all arithmetic means between 1 and 9. You will not normally be asked to identify arithmetic means, you will be asked to find arithmetic means. Let's look at some examples:

Example 1

Find one arithmetic mean between 2 and 16.

Solution

In order to develop a plan, lets draw a simple diagram of what we are looking for. We want to put ONE number between 2 and 16 so that the result is an arithmetic progression. Look:

2, ____, 16

Your job is to fill in the blank with the only number that works. You could probably find this number by guessing and checking, but you should develop a strategy that will help with the more complex examples.

Now we can use 'The Formula' to find the common difference for this A.P. We know that = 2 and = 16. For the terms we are examining, n = 3. Fill in the formula for what you know, and solve for d:

Knowing that the common difference of this A.P. is 7, it should be straightforward to fill in the arithmetic mean. It is 2 + 7 = 9. That makes this little progression:

2, 9, 16 ...

Example 2

Place three arithmetic means between 4 and 52.

Solution

Again, let's look at what we need to do. We need to place 3 numbers between 4 and 52 so that the result is an arithmetic progression. This would look like:

4, ___, ___, ___, 52 ...

All of these 'in-between' numbers are considered arithmetic means. To find the answer, we need to know the common difference for this A.P. We know that = 4and = 52. So using an index number (n) of 5, we get the following work:

Knowing that the common difference is 12, we can fill in all the 'in-between' numbers to achieve our answer:

4, 16, 28, 40, 52

Now that you have seen some examples involving arithmetic mean problems, you can try some sample questions or head back to the Arithmetic Progressions main page.