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

##### 1 Answer

May 8, 2016

#### Explanation:

The sum is: *number of terms* *average term*.

The number of terms in our example is

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

#(1+100)/2 = 101/2#

So:

#1+2+...+99+100 = 100xx(1+100)/2 = 50xx101 = 5050#

Another way of looking at it is:

#1+2+...+99+100#

#={:(color(white)(00)1+color(white)(00)2+...+color(white)(0)49+color(white)(0)50+), (100+color(white)(0)99+...+color(white)(0)52+color(white)(0)51) :}#

#={: underbrace(101+101+...+101+101)_"50 times" :}#

#=101xx50 = 5050#