Today is another Fibonacci day. It is the 3rd, which is a Fibonacci number. We've looked at a bunch of addition patterns in the Fibonacci numbers, like adding them, or the even ones. What about the odds? How about we look.
1 = 1
1 + 2 = 3
1 + 2 + 5 = 8
1 + 2 + 5 + 13 = 21
See the pattern? We are getting the Fibonacci numbers! Why? Let's look at each one as the previous two.
1 + (1 + 1) + (2 + 3) + (5 + 8)
You are basically adding the Fibonacci numbers up, and then adding one! that's a Fibonacci number minus one plus one, or a plain Fibonacci number.
Bonus: We've done quite a bit with explicit formulas lately. I think now is the time to mention the explicit formula for Fibonacci numbers. For the nth Fibonacci number, you do:
[1/(5^.5)](φ^n - φ^n)
If you don't know, the Greek letter fi means the golden ratio, or 1.618... Fi with a bar on top is what you get with a small change in the golden ratio's formula, -0.618...
This is definitely complicated, but I couldn't believe that worked. I checked many numbers just to convince myself it worked!!