It seems to me that mathematical explanations and proofs should, whenever possible, relate to our intuition. Here is an example of what I mean. It is a little long, but I hope it is easy to follow.

Suppose we want to find a formula for the sum of the numbers from 1 to n. Let’s consider n=5 and write the numbers: 1 2 3 4 5.

Now let’s ask a simpler question. What would you guess is the average of those numbers? I am hoping that, without much hesitation, you would say 3, the one in the middle. From the average, it is easy to get the sum. We just multiply by 5 to get 15, and adding the numbers we do indeed get 15.

Let’s suppose for now that we can generalize this to all numbers n. How do we find the value in the middle without writing down all the numbers? That is easy. It is the average of the first and the last, (n+1)/2. We therefore get the sum by multiplying by n to get n(n+1)/2.

Let’s now use what we have found to come up with a formal derivation. We note that not only do the first and last add up to n+1, but so the the second and next to last (2 and n-1), the third and third to last (3 and n-2), and so on. Each time one number goes up by 1 in value and the other goes down by 1. If we write the numbers 1 to n and underneath write the numbers from n down to 1, we get n pairs of numbers adding to n+1. Therefore twice the sum of the numbers is n(n+1) and the sum must be half of that, or n(n+1)/2.