Arithmetic Sequence Formula: Arithmetic Mean

We consider the following arithmetic sequence:

It is clear that and are two terms that are located at the same distance from . ( on the right and on the left).

Let's write using the general term formula:

And let's write using the general term formula:

Now, we add the formulas (1) and (2) side by side:

And consequently:

This formula is called the mean value of an arithmetic sequence.

This means that the term is located exactly in the middle between the two terms and .

Note:

The mean value of an arithmetic sequence is:

As a result of the above, the sequence is an arithmetic sequence if and only if


Here are some examples:

Example1:

Let three consecutive numbers in an arithmetic sequence. Let's calculate .

Solution:

we've got:

Using the mean arithmetic formula:


Example2:

Find the general term of an arithmetic sequence such that and .

Solution:

The term 13 is located on the same distance between the terms and , so:

Let's calculate

Then, the general term is: