# What is the sum of the first 100 consecutive positive integers?

May 8, 2016

$5050$

#### Explanation:

The sum is: number of terms $\times$ average term.

The number of terms in our example is $100$

The average term is the same as the average of the first and last term (since this is an arithmetic sequence), namely:

$\frac{1 + 100}{2} = \frac{101}{2}$

So:

$1 + 2 + \ldots + 99 + 100 = 100 \times \frac{1 + 100}{2} = 50 \times 101 = 5050$

Another way of looking at it is:

$1 + 2 + \ldots + 99 + 100$

$= \left.\begin{matrix}\textcolor{w h i t e}{00} 1 + \textcolor{w h i t e}{00} 2 + \ldots + \textcolor{w h i t e}{0} 49 + \textcolor{w h i t e}{0} 50 + \\ 100 + \textcolor{w h i t e}{0} 99 + \ldots + \textcolor{w h i t e}{0} 52 + \textcolor{w h i t e}{0} 51\end{matrix}\right.$

$= \left.{\underbrace{101 + 101 + \ldots + 101 + 101}}_{\text{50 times}}\right.$

$= 101 \times 50 = 5050$