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Arithmetic Series

An arithmetic series is a series whose related sequence is arithmetic. It results from adding the terms of an arithmetic sequence .

Example 1:

Finite arithmetic sequence: $5, 10, 15, 20, 25, ..., 200$

Related finite arithmetic series: $5 + 10 + 15 + 20 + 25 + ... + 200$

Written in sigma notation: $\sum_{k = 1}^{40} 5 k$

Example 2:

Infinite arithmetic sequence: $3, 7, 11, 15, 19, ...$

Related infinite arithmetic series: $3 + 7 + 11 + 15 + 19 + ...$

Written in sigma notation: $\sum_{n = 1}^{\infty} (4 n - 1)$

To find the sum of the first $n$ terms of an arithmetic sequence, use the formula
$S_{n} = \frac{n (a_{1} + a_{n})}{2}$ ,
where $n$ is the number of terms, $a_{1}$ is the first term, and $a_{n}$ is the last term.

Example 3:

Find the sum of the first 20 terms of the arithmetic series if $a_{1} = 5$ and $a_{20} = 62$ .

$\begin{array}{l} S_{10} = \frac{20 (5 + 62)}{2} \\ = 670 \end{array}$

Example 4:

Find the sum of the first $40$ terms of the arithmetic series
$2 + 5 + 8 + 11 + ...$ .

First find the $40^{th}$ term:

$\begin{array}{l} a_{40} = a_{1} + (n - 1) d \\ = 2 + 39 (3) \\ = 119 \end{array}$

Then find the sum:

$\begin{array}{l} S_{n} = \frac{n (a_{1} + a_{n})}{2} \\ S_{10} = \frac{40 (2 + 119)}{2} \\ = 2420 \end{array}$

Example 5:

Find the sum:

$\sum_{k = 1}^{50} (3 k + 2)$

First find $a_{1}$ and $a_{50}$ :

$\begin{array}{l} a_{1} = 3 (1) + 2 = 5 \\ a_{50} = 3 (50) + 2 = 152 \end{array}$

Then find the sum:

$\begin{array}{l} S_{k} = \frac{n (a_{1} + a_{k})}{2} \\ S_{50} = \frac{50 (5 + 152)}{2} \\ = 3925 \end{array}$