Finding Sums of Finite Arithmetic Series

A series is a summation of the terms in a sequence. For the sequence $2,$ $4,$ $6,$ $8,$ the series can be written as $$2+4+6+8.$$ If the terms are given by a rule, it's usually more compact to write it using sigma notation. The sequence above is described by the explicit rule $a_n=2n.$ The series can be written as $$\sum _{n=1}^{4}2n.$$ Here, by substituting $n$ with integers $1$ (under $\Sigma$) through $4$ (above $\Sigma$) the individual terms of the series are obtained, and the sigma itself indicates that between each term is a plus sign. This alternative notation is useful if the series contains a large number of terms, and especially useful when the sequence is infinite. Then, the infinity symbol is written above the sigma.

The summation of the terms in an arithmetic sequence is called an arithmetic series. If the sequence is short enough, such as $1,3,5,7,$ it's straightforward to calculate the sum: $$s=1+3+5+7=16.$$