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:
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 .
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:
Let three consecutive numbers in an arithmetic sequence. Let's calculate .
Using the mean arithmetic formula:
Find the general term of an arithmetic sequence such that and .
The term 13 is located on the same distance between the terms and , so:
Then, the general term is: