The Gauss triangle trick

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.

Theorem: \sum_{k=1}^nk=\frac{n(n+1)}{2}.

Proof: Let S=1+2+\cdots n and write 2S as follows:

1+2+\cdots n +\\n+(n-1)+\cdots 1.

Now addition is associative, so adding each number in the top line with the number directly below it, we get 2S= (1+n) + (2+(n-1))+\cdots (n+1). Clearly there are n parenthesis with each summing to n+1 and so 2S=n(n+1). The  result follows.\Box

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 1 to 100. 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 1,1+2=3,1+2+3=6 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.

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