Dots, numbers and the mind of a child genius, part 2

In part 1, we saw how Gauss, at the age of 8, found a quick way to add all the counting numbers from 1 to 100. The question that was left hanging was, what if the teacher had asked him to stop at 99? After all, his trick of pairing the numbers (100 + 1, 99 + 2, ...,  51+50) could not work for an odd number of values. Or could it?

Well, the intuition here is simple: Adding zero does not change a sum. Using this fact, we can change our question to: What is the sum of the whole numbers from 0 to 99?

This reduces the sum to a simple calculation: 50 groups of 99 or 

A visual representation of the problem

Let's start small and answer the question, "What is the sum of the counting numbers from 1 to 9?"

We can go back to where it all started: Representing our numbers with dots in a triangular formation. This time, we will arrange the dots in a right-angled triangle:

Please don't add up the dots! Let's just notice that the blue dots, which we want to total, are half of the dots of the following, 10 by 9,  rectangle. 

Now, we can answer the question, what is the sum of the counting numbers from 1 to 99, by imagining a right-angled triangle with 1 dot at the top and 99 dots at the bottom. The number of these dots will be half of the number of dots in a 100 by 99 rectangle:

Let's generalise

As hobbyist mathematicians, we are not interested in the sum of counting numbers from 1 to 100 or 1 to 99, specifically. We are interested in the general case: What is the sum of the counting numbers from 1 to any other number we wish to stop at?

Let's call the top number we want to reach n. The triangle will have one dot at the top and n dots at the bottom. The total number of dots will be half of the dots in an n by (n + 1) rectangle.

For those who want it stated in the way they would see it in a maths class: