Man’s mind, once stretched by a new idea, never regains its original dimensions.
— Oliver Wendell Holmes
As you probably know, I wear my heart on my sleeve:
Well, I took the golden opportunity (ha!) to bring the golden ratio into Calc 2 this week, using it (and its little pal ) to find a closed formula for the -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:
And let’s say for some reason, you need to cook up . 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 -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.