An arithmetic series is the sum of the terms in an arithmetic sequence with a definite number of terms. Following is a simple formula for finding the sum:

**Formula 1:**

If \(S_{n}\) represents the sum of an arithmetic sequence with terms \(a_{1}, a_{2}, a_{3}, \ldots a_{n}\), then \(S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)\)

This formula requires the values of the first and last terms and the number of terms.

Since \(a_{n}=a_{1}+(n+1) d\)

Then \(a_{1}+a_{2}=a_{1}+\left[a_{1}+(n-1) d\right]\)

\(a_{1}+a_{n}=2 a_{1}+(n-1) d\)

Substituting this last expression for \(\left(a_{1}+a_{n}\right)\) into Formula 1 , another formula for the sum of an arithmetic sequence is formed.

**Formula 2:**

\(S_{n}=\frac{n}{2}\left[2 a_{1}+(n-1) d\right]\)

This formula for the sum of an arithmetic sequence requires the first term, the common difference, and the number of terms.