Arithmetic Series
Sum arithmetic sequences using the first term, last term and common difference.
An arithmetic series is the sum of the terms in an arithmetic sequence.
Sum of the first \(n\) terms\[S_n=\frac{n}{2}(u_1+u_n)\qquad\text{or}\qquad S_n=\frac{n}{2}\left(2u_1+(n-1)d\right)\]
Worked example
Find \(S_{25}\) for \(3, 8, 13, 18,\ldots\).
Here \(u_1=3\), \(d=5\), and \(n=25\).
\[S_{25}=\frac{25}{2}\left(2(3)+24(5)\right)=\frac{25}{2}(126)=1575\]When to use each formula
- Use \(S_n=\frac n2(u_1+u_n)\) when you know the first and last term.
- Use \(S_n=\frac n2(2u_1+(n-1)d)\) when you know the first term and common difference.