Daily Archives: April 15, 2014

Golden Power Series!

As you probably know, I wear my heart on my sleeve:

Well, I took the golden opportunity (ha!) to bring the golden ratio \Phi = \frac{1+\sqrt{5}}{2} into Calc 2 this week, using it (and its little pal \Psi = \frac{1-\sqrt{5}}{2}) to find a closed formula for the n-th term of the Fibonacci sequence.

The ubiquitous Fibonacci sequence! It’s something you may have encountered out in the wild. You know, it goes a little like this:

F_0 = 1, \, \, F_1 = 1, \, \, F_n = F_{n-1} + F_{n-2},
so F_2 = 2, \, \, F_3 = 3, \, \, F_4 = 5, \, \, F_5 = 8, \, \, F_6 = 13, \, \, F_7 = 21 \, \, \ldots.

And let’s say for some reason, you need to cook up F_{108}. I hope you have some time on your hands if you’re planning to add all the way up to that. Instead, wouldn’t it be nice if we had a simple formula that we could use — i.e., a formula that was not recursive — to figure out the n-th Fibonacci number?

Luckily, such a formula exists, and there are lots of ways to find it. In this post, we’ll find it using power series. Read on, brave blogosphere traveler.