The Sum of the First n Positive Integers

According to an anecdote, the math genius Carl Friedrich Gauss, added all the numbers from 1 through 100 when he was at a very young age. The sum of the first positive 100 integers is 5050 and Gauss was able to give this sum in less than a minute. True or not, how can we get the sum of all these numbers with admirable speed? Well, it’s not really that hard.

sum-of-first-n-positive-integers

If you want to be a great mathematician someday like Gauss, you must learn to explore problems and look for patterns. Notice that if you add the first $latex 100$ integers, the sum of the largest and the smallest integer is $latex 101$. Also, the sum of the second largest and the second smallest is $latex 101$. This also goes with the third largest and the third smallest and so on. Since there are the first $latex 100$ numbers can be divided into $latex \frac{100}{2}=50$ pairs, each of which has a sum of $latex 101$, the sum of all the numbers is $latex 50(101) = 5050$. We will use this strategy to generalize. We explore the sum of the first positive n integers. Continue reading