Fibonacci Numbers and the Golden RatioLesson Plans > Mathematics > Algebra > Functions
Fibonacci Numbers and the Golden Ratio
The answer to this question is a sequence of numbers known as the 'Fibonacci sequence':
These numbers can be described mathematically as follows:
F(n+2) = F(n+1) + F(n)
In other words, each Fibonacci number is the sum of the two preceding Fibonacci numbers. Another way of saying that is, If you know two successive Fibonacci numbers, you can find the next one by adding them together.
There are many reasons why Fibonacci numbers are interesting to mathemeticians. One of the reasons is that Fibonacci numbers are very closely related to the Golden Ratio, a number which the ancient Greeks (and math nerds like myself) find very interesting. Here are three interesting facts about Fibonacci sequences and the Golden Ratio. Advanced students may be interested in trying to prove these statements.
- F(n) =
- As n approaches infinity, approaches the limit pF(n)
- If a Fibonacci sequence is also a geometric sequence, its ratio is either p or p'.
A research project for students:
Fibonacci numbers appear surprisingly often in the natural world (aside from the hypothetical rabbit problem at the beginning of this article) Numbers of spirals in different kinds of pine cones, thorn bushes, and sunflowers are often Fibonacci numbers. How many natural occurances of Fibonacci numbers can you find?