What is an arithmetic series?

An arithmetic series is a series of an arithmetic sequence. Given an arithmetic sequence equation whose equationth term is equation, we can define its arithmetic series equation (which is the sum of the first equation terms of the sequence) as:

    equation

According to the above formula, we only need the first term equation and the equationth term equation to find equation.

Another equation for the series is:

    equation

According to this formula, we need the first term equation and the difference equation to find equation.

Example of an arithmetic series

Consider the following sequence:

    equation

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 equation and we only need to find the difference equation to be able to use the formula. We find equation as the difference of any two successive terms: equation. We obtain the series equation as:

    equation

Now we can easily find the sum of the first 100 terms as equation.