Gauss: Sum from 1 to 100 trick

Interesting math trick. I wanted to relearn it. What if you wanted to add all the integers from 1 to 100?

S being the sum of integers from 1 to 100

S = 1 + 2 + 3 + 4 + ... + 98 + 99 + 100

We can rewrite the sum like this:

S = (100+1)+(99+2)+(98+3)+...+(53+48)+(52+49)+(51+50)

Then, we realize that 100+1=99+2=98+3=...=(51+50) = 101

We know there will be 50 pairs of 101, so the sum of all integers from 1 to 100 is 50*101 or 5050.

In general what we are saying is that the sum integers from 1 to some greater integer, N, is the product of 1 more than N (e.g. 100 in the case above) and half N.

S = (N/2)*(N+1)

example from 1 to 10

S = (10/2)*(10+1) = 55

http://www.wolframalpha.com/input/?i=1%2B2%2B3%2B4%2B5%2B6%2B7%2B8%2B9%2B10

This math trick is interesting because it was supposedly invented/discovered by a great mathematician when he was a child, Carl Friedrich Gauss.