Arithmetic Series

Example 1:

Finite arithmetic sequence: $5,10,15,20,25,\mathrm{...},200$

Related finite arithmetic series: $5+10+15+20+25+\mathrm{...}+200$

Written in sigma notation: ${\displaystyle \underset{k=1}{\overset{40}{\sum }}5k}$

Example 2:

Infinite arithmetic sequence: $3,7,11,15,19,\mathrm{...}$

Related infinite arithmetic series: $3+7+11+15+19+\mathrm{...}$

Written in sigma notation: ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\left(4n-1\right)}$

Example 3:

Find the sum of the first 20 terms of the arithmetic series if $a_1=5$ and $a_{20}=62$ .

$\begin{array}{l}{S}_{10}=\frac{20\left(5+62\right)}{2}\\ =670\end{array}$

Example 4:

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

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

$\begin{array}{l}{a}_{40}=a_1+\left(n-1\right)d\\ =2+39\left(3\right)\\ =119\end{array}$

Then find the sum:

$\begin{array}{l}{S}_n=\frac{n\left(a_1+a_n\right)}{2}\\ S_{10}=\frac{40\left(2+119\right)}{2}\\ =2420\end{array}$

Example 5:

Find the sum:

${\displaystyle \underset{k=1}{\overset{50}{\sum }}\left(3k+2\right)}$

First find $a_1$ and $a_{50}$ :

$\begin{array}{l}{a}_1=3\left(1\right)+2=5\\ a_{50}=3\left(50\right)+2=152\end{array}$

Then find the sum:

$\begin{array}{l}{S}_k=\frac{n\left(a_1+a_k\right)}{2}\\ S_{50}=\frac{50\left(5+152\right)}{2}\\ =3925\end{array}$