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What is an arithmetic series?
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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.

by Diamond (66,975 points)

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