Question

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

Answer 1

`5050`

Explanation:

The sum is: number of terms `xx` 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:

`(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`