This is a part of a series of proofs I plan to write here which seem essential for any student of mathematics. This series is inspired by the stackexchange post here, although I am not sure yet as to how many proofs from the list I will pick up.
Proof: Let and write as follows:
Now addition is associative, so adding each number in the top line with the number directly below it, we get . Clearly there are parenthesis with each summing to and so . The result follows.
So why is this called the Gauss triangle trick. Well, the story goes that when Gauss was in preparatory school, Gauss’s teacher told the kids to add up all the numbers from to . He probably thought that this will give him some time to rest and was surprised when in a few minutes Gauss came up with answer using this trick.
The numbers obtained as the various sums, eg etc are called triangle numbers. A triangle number is so called because it is a count of the number of balls that can form a equilateral triangle.