Fibonacci Day: Addition of Fibonacci Numbers

As you may have noticed, today is a Fibonacci Day. If you'll notice, it is the second, and two is a Fibonacci Number. 

Let's look at a simple pattern within these numbers. What would happen if you add all the Fibonacci numbers up? Infinity, because they go on forever. What if you added Fibonacci numbers, and then stopped at some point. Let's see:

1 = 1
1 + 1 = 2
1 + 1 + 2 = 3
1 + 1 + 2 + 3 = 7
1 + 1 + 2 + 3 + 5 = 12

You might not see a pattern, but there is one. Let's rewrite these sums in a different format.

1 = 1 = (2 - 1)
1 + 1 = 2 = (3 - 1)
1 + 1 + 2 = 4 = (5 - 1)
1 + 1 + 2 + 3 = 7 = (8 - 1)
1 + 1 + 2 + 3 + 5 = 12 = (13 - 1)

See it now? Every sum is one less than a Fibonacci number! This pattern will actually go on forever! In order to prove it, we will use something called proof by induction. 

If we were to add on the next Fibonacci Number (21), we would also be adding that to the 13 - 1 from before. However, with 13 and 21 being consecutive Fibonacci Numbers, they join together to create the next Fibonacci Number. Since the minus one remains, you always are subtracting one from a Fibonacci Number. 

For another proof, let's express each Fibonacci Number as the difference of the two after it. 

(2-1) + (3-2) + (5-3) + (8-5) + (13-8) + (21-13)

You'll see that in the first two expressions, the 2 and -2 cancel out. In the next two, the 3 and -3 cancel. This keeps going until you are left with the greater number in the last expression and the -1 from the first one. That is also a very beautiful proof. Isn't that cool?!