Sum of an Arithmetic Series

An Arithmetic Series is a sum over a sequence of numbers starting at $a_0$ that jump by a common difference $d$ and that have $N$ terms

$$\begin{array}{rl}\sum _{i=0}^n{a}_0+d_i& =N(\frac{\text{first}+\text{last}}{2})\\ & =(n+1)(\frac{a_0+(a_0+d_n)}{2})\end{array}$$

Process

1. Add in another copy of the sum and reverse the direction.
2. Group each term together to sum to first + last
3. There are $n$ pairs

$$\begin{array}{rl}1+2+...+(n-1)+(n)& =\sum _{k=1}^{n}k\\ n+(n-1)+...+2+1& =\sum _{k=1}^{n}k\\ (n+1)+(n+1)+...+(n+1)+(n+1)& =2\sum _{k=1}^{n}k\\ (n+1)n& =2\sum _{k=1}^{n}k\\ \frac{n(n+1)}{2}& =\sum _{k=1}^{n}k\end{array}$$

Example

Examples	First	Last	$d$	$N$	Sum
$1+2+3+4+5+6+7+8+9+10$	1	10	1	10	$\frac{11\cdot 10}{2}$
$1+3+5+7+9+11+13+15$	1	15	2	8	$\frac{16\cdot 8}{2}$
$21+31+41+51+61+71+81+91+101$	21	101	10	9	$\frac{122\cdot 9}{2}$