What is an arithmetic series?
An arithmetic series is a series of an arithmetic sequence. Given an arithmetic sequence whose th term is , we can define its arithmetic series (which is the sum of the first terms of the sequence) as:
According to the above formula, we only need the first term and the th term to find .
Another equation for the series is:
According to this formula, we need the first term and the difference to find .
Example of an arithmetic series
Consider the following sequence:
Let us find the sum of the first 100 terms. Of course we could determine all 100 terms and then sum them up. Or we can use the above formula. We already know the first term and we only need to find the difference to be able to use the formula. We find as the difference of any two successive terms: . We obtain the series as:
Now we can easily find the sum of the first 100 terms as .