For a full introduction to this series, see my earlier article. Briefly, in this series I show interesting ideas in math that can be understood with only the math people learn in elementary school.
A story about Gauss
Carl Friedrich Gauss (1777-1855) would be on anyone’s list of the greatest mathematicians of all time. One early sign that he was something special is illustrated by the following story. His teacher assigned the class to add up the numbers from 1 to 100. Why he did this varies depending on the version of the story (in one version he was suffering from a hangover; in another he was just an ogre, etc). Almost as soon as he was done stating the problem, Gauss wrote the answer on his slate. The habit in Germany at the time was to wait for each student to complete the assignment. After a long wait, the teacher looked at the answers. Only Gauss’ answer was correct.
Suppose you were assigned to add up the numbers from 1 to 100 and weren’t allowed to use a calculator. You have the advantage that you know there is a method to do so quickly. What might you try? Well, of course, you could try just adding them. But you’d likely make a mistake and it would certainly take a while. Let’s try playing instead. Go ahead and play around a bit before reading on.
What Gauss did
Gauss noticed something. He noticed that 1 + 2 + 3 + …. + 99 + 100 = 1 + 100 + 2 + 99 …. = 101 + 101 + … + 101. Since there were 50 pairs, the answer is 101*50 = 5050.
Turning the answer into a formula
It’s one thing to have this insight and another to turn the insight into a formula. To figure out what the formula is, we first figure out how many pairs there are. When there are an even number of numbers (as in 1 to 100) there are n/2 pairs (with 1 to 100 there are 50). What about with an odd number of numbers? There can’t be n/2 pairs can there? Yes there can! Let’s look at a case with a small number, say 5. 1 + 2 + 3 + 4 + 5 = 1 + 5 + 2 + 4 + 3 = 6 + 6 + 3. That 3 is just half of the 6! Cool!. So, we start with n/2.
The second thing to do is figure out what n/2 needs to be multiplied by. With n = 100 it was 101. Hmm. Just 1 more. Is that always true? Try with some other numbers. Play a bit. It’s always true. So, the formula is n*(n+1)/2. For n = 100 this is 100*101/2 = 5050.
Try other series! What about adding only odd numbers? Only even numbers? A series that starts with something other than 1? I’ll cover some of these in later articles. But, in the meantime, PLAY!