Sum of an Arithmetic Series. How easy is it to add up the numbers $1$ to $100$? $1$ to $1000$? $1$ to $1\,000\,000$?

How easy is it to add up the numbers $1$ to $100$? $1$ to $1000$? $1$ to $1\,000\,000$?

Example: Add up the numbers from $1$ to $100$.
There is a nice way to do this by writing the sum in two ways, adding them up and dividing by $2$.

$$\begin{array}{llllll} 1&+2&+3&+4&+\cdots &+100\\ 100&+99&+98&+97&+\cdots &+ 1\\\hline 101&+101&+101&+101&+\cdots &+ 101 \end{array}$$

Noting that $101$ occurs $100$ times these add up to $100\times 101$.

Hence the sum

$$1+2+3+4+\cdots +100=\displaystyle {100\times 101 \over 2}$$

Did you notice that the first and last terms of the series add up to $101$. This is how the general case works too.

Sum of an Arithmetic Series

The sum of the arithmetic series with $n$ terms

$$A+ (A+d) + (A+2d)+\cdots + L$$

is

$$S_n={n(A+L)\over 2}$$

where

$A=$ the first term

$L=$ the last term

$n=$ the number of terms in the series

$L=$ the last term

For the example above, 
$A=1$, $L=100$ and $n=100$ (and $d=1$ but we didn't need it here).