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Fibonacci Numbers and the Golden Ratio

Lesson Plans > Mathematics > Algebra > Functions
 

Fibonacci Numbers and the Golden Ratio

Suppose you have a pair of rabbits given to you as babies. This pair of rabbits will begin to reproduce when they are two years old. They will produce another pair of rabbits every year. Each of these pairs of rabbits will also reproduce--one rabbit pair per year starting when they are two years old. How many rabbit pairs will you have after n years?

The answer to this question is a sequence of numbers known as the 'Fibonacci sequence':

1,1,2,3,5,8,13,...

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) =
    pn - p'n
    5
  • As n approaches infinity,
    F(n+1)
    F(n)
    approaches the limit p
  • If a Fibonacci sequence is also a geometric sequence, its ratio is either p or p'.

For the above formulas,

p = The Golden Ratio =
1 +
5
2


p' =
1 -
5
2



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?

 

Lesson by Mr. Twitchell

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