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.

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 integers, the sum of the largest and the smallest integer is . Also, the sum of the second largest and the second smallest is . This also goes with the third largest and the third smallest and so on. Since there are the first numbers can be divided into pairs, each of which has a sum of , the sum of all the numbers is . We will use this strategy to generalize. We explore the sum of the first positive n integers.

In adding the first integers, we have two cases. The first case, if the largest integer is even, like , and the second case is if the largest is odd.

Case 1: Largest integer is even.

If the largest integer is even, we can pair the numbers as shown below.  We can pair and , and , and , and so on.

Notice that each sum is equal to and there are pairs. Therefore , the sum of all the positive integers from to is

.

Case 2:  Largest integer is odd

If the largest integer is odd, we can only pair integers.  We can pair and , and and and and so on.

The sum of each pair above is and there are pairs. So the sum of the first positive integers is . But do not forget that we still have to add the largest digit which is .  So, the sum of all the numbers from to where is odd is

As we can see, in any case, the sum of the first positive integers is .

The generalization above is not considered as a proof of the sum of all integers. A formal proof called mathematical induction is needed to show that it is true to all positive integers.