First a little pre-history. Back in 1784 (I think this was the year) one lazy math teacher in some remote German city wanted to occupy his class for the duration of an entire lesson. He gave the class the following assignment: to add all the consecutive numbers from 1 to 100. The teacher thought that it would take a class an entire lesson to do this and started to read a newspaper. (I actually don’t know for sure, I am just guessing.)
However, after five minutes of reading one of his pupils interrupted him, claiming that he’s already done the task and give him the correct answer of 5050. The teacher was surprised and the pupil to show him the solution. When he saw it, he was amazed even more, because it was so simple and beautiful. The little boy grew out to be a great mathematician known to the world as Carl Friedrich Gauss. Below is his brilliantly simple solution.
Let’s take series of numbers:
0,1,2,3,4,5…49,50,51…95,96,97,98,99,100
Now duplicate this series
0,1,2,3,4,5…49,50,51…95,96,97,98,99,100
0,1,2,3,4,5…49,50,51…95,96,97,98,99,100
Then reverse the second row and superimpose it with the first one in the following fashion
Notice that the sum of each column would be 100. All we have to do is to calculate how many of them.
That’s 1 to 99 gives us 99 x 100 plus two 100’s on both ends. That will be 9900+200 = 10100
But since we added a second set at the beginning, we need to divide the result by 2
10000/2 = 5050
That's all folks!