When I was in 9th 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 10th 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 12th 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 n2/2
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 n2/2 + 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.