### It's as easy as 1, 2, 3

There can hardly be anything simpler than learning to count. Most of us have mastered it before we started elementary school. Similarly we all knew our a-b-c's before we graduated kindergarten. Yet any attempt to combine the two, leading to the dreaded realm of algebra, and most people's brains turn into jello, or so they say. Well, it doesn't have to be that way, and if you keep an open mind and read this article through to the end, you may discover that algebra is not only easy, but fun too.

Let's start with something simple, like counting. One, two, three, four, five ... not so hard is it. Suppose we start adding these numbers together, as follows:
1 = 1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15

### Oh no, Algebra!

By using a letter instead of number. Let's use the letter N. So we would ask, what is the sum of the following sequence: 1 + 2 + 3 + 4 + ... + N? Can it possibly be that simple? Yes! All we have to remember is that N stands for a number. It could be 5, it could be 100, it could be anything. Now what is the number that is one less than N? Why N-1 of course. How about the number that is 2 less than N? Try N-2 on for size. Let's rewrite the sequence now adding in a few more terms on the end:
1 + 2 + 3 + 4 ... + (N-2) + (N-1) + N = ?
All I've done is explicitly show you the last few terms of the sum "symbolically." Instead of ending with 98 + 99 + 100 or 998 + 999 + 1000, I've used the letter N to represent the last number. Now let's apply the profound notion that 2 + 3 = 3 + 2 and rearrange the sequence like Gauss did. We get
(1 + N) + (2 + N-1) + (3 + N-2) + ...
We can write (2 + N - 1) as (N + 2 - 1) = (N+1). Also we can write (3 + N - 2) as (N + 3 - 2) = (N+1). Each of these terms adds up to N+1, and there are exactly N/2 of them, so what is the total? Well, it must be (N+1) times N/2, or if we write it the way mathematicians do, N*(N+1)/2 Now plug a number for N into this expression, and we have the sum of the first N numbers. If N is 5 you get 5 * 6 / 2 = 15 correct? If N is 100 you get 100*101/2 = 50*101 = 5,050 just like Gauss. It is a lot easier to calculate N*(N+1)/2 than it is to add up N numbers, yet the answer is always the same. This idea of letting letters stand for numbers is the heart of algebra, and allows us to express very complicated ideas in a very simple way. Now why didn't they tell you it was that easy in high school?