Just Add the Numbers? Roman Kvasov
Key Problem Find the sum: 1 + 2 + 3 + … + 98 + 99 + 100
Solution Finding this sum is a boring and unpleasant process, where even a little error might lead to a result very different from the real solution. It means that to attack this problem we will have to invent some trick to find the sum in an easier way. Note that 1 + 100 = 101, 2 + 99 = 101, and 3 + 98 = 101. So, if instead of adding all the numbers in the order they appear, we combine them in a clever way adding the first with the last and so on, so that we will have to just add 101’s : 1 + 2 + 3 + … + 98 + 99 + 100 = (1 + 100 ) + (2 + 99 ) + (3 + 98 ) + … + (50 + 51 ) = (101) + (101) + (101) + … + (101) = 101 × 50 = 5050
Conclusion In order to add numbers from 1 to some number n, do the following • • •
take this number n and add 1 to it multiply the result by the number n divide the result by 2
Following this idea for the problem discussed above, we take 100, add one to it, multiply it by 100 and divide it by 2: (100 + 1) × 100 ÷ 2 = 101 × 100 ÷ 2 = 10100 ÷ 2 = 5050