# Area of triangle & sum of n natural numbers

When I was in 9^{th} grade, I fell really ill with Typhoid. As a result I had missed many months of attending school. During this time, I was selected for an interview for the Kishore Vaigyanik Protsahan Yojana scholarship. To my joy, I had qualified the written prelims and had advanced to the final interview stage.

During the interview I was asked a single question - what is the formula for the sum of first *n* natural numbers. Now in India, they teach you about Arithmetic Progression only in the 10^{th} grade so I had no idea about what the actual answer would be. They told me that it was *n(n+1)/2*. When I saw this, I imagined an isoceles right angled triagle with sides of size *n* and the formula was very close to its area. I told them that but the extra *n/2* threw me off. I didn’t pass that interview.

Years later, in 12^{th} grade I learnt about integration in calculus and that’s when it hit me that I had the right idea. The integral is indeed the area under the graph. So in my right angled triangle visualization, the formala was actually the area of under a line that represented the hypoteneuse of a right angled traingle.

∫n from *0* to *n* is actually n^{2}/2

But we are still missing the extra *n/2*. This is explained by observing that for every pixel on the hypoteneuse of the triagle, we are missing half a square in the computed area. The hypoteneuse is made up of exactly *n* pixels and the area of the half-pixels lying on the hypoteneuse is *n/2*.

So the sum of natural numbers is the area under the triangle *n ^{2}/2* + the area of half pixels on the hypoteneuse

*n/2*which is the same as the well-known formula for the sum of first n natural numbers,

*n(n+1)/2*.