Gauss and Arithmetic Progression

I have just done a mathematical problem in my work. Interestingly, it brings back fond memories of those days when I was fascinated by stories of brilliant mathematicians.

What struck me is that while there are many ways to tackle a problem, if we could find a formulae, we would cut down the time needed to reach the same conclusion. In fact, the formulae could be a very simple - that's the elegance of Mathematics.

One of which was the many fascinating stories of how mathematician solved seemingly difficult questions. This is my favorite about Gauss.

Gauss was a famous German mathematican (1777 to 1855). He was a child prodigy. There were many stories about his precocity. One was when he was in primary school and his teacher wanted to occupy his students by asking them to sum up a list of integers - from 1 to 100.

But Gauss did it in split second. How did he do it?

This is how:

To find, Sum = 1 + 2 + 3 + ..... + 99 + 100 [There are 100 terms here]
Now Sum is also = 100 + 99 + 98 +.... + 2 + 1

So Sum + Sum = 101 + 101 + ...........+ 101 + 101 [There are 100 101s here]
2 Sum = 10,100
Sum = 10100/2
= 5,050 [Whoala!]

To this day, I never memorise the formula for arithmetic sum but always derive it from first principle. You can use similar principle to get the formula for the sum of geometric progression. Try it :)