The Proof Behind the Sum of the First n Natural Numbers

Understanding the sum of the first n natural numbers is a fundamental concept in mathematics. The formula 1+2+3+…+n=n(n+1)/2 provides a quick way to calculate this sum. In this article, we will explore the reasoning behind this formula through a simple yet elegant proof.

Let’s consider the sum of the first n natural numbers as follows:

The sum of the first n natural numbers

Now, let’s rearrange the terms in this sum in two different orders. First, let’s write S in its original order. Secondly, let’s reverse the order of the terms:

Original and reverse order of summation

Now, let’s add these two expressions of S term by term:

When both sides of the equation are added together

In the expression on the right-hand side, (n+1) appears n times because there are n terms in both the original and reversed sums. So, we can simplify the equation as:

Dividing both sides by 2, we get:

Therefore, we have successfully proved that the sum of the first n natural numbers can be calculated using the formula 1+2+3+…+n=n(n+1)/2. This proof provides valuable insight into the relationship between consecutive natural numbers and is a fundamental concept in mathematics.