Applications of the Fibonacci sequence The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any other reason you would care about the Fibonacci sequence?
Perhaps it's not an entirely practical application, but Fibonacci numbers can be used to convert from miles to kilometers and vice versa:


  Take two consecutive Fibonacci
  numbers, for example 5 and 8. And
  you're done converting. No kidding –
  there are 8 kilometers in 5 miles. To
  convert back just read the result from
  the other end - there are 5 miles in 8
  km!


But why does it work?


  Fibonacci numbers have a property that
  the ratio of two consecutive numbers
  tends to the Golden ratio as
  numbers get bigger and bigger. The
  Golden ratio is a number and it
  happens to be approximately 1.618.
  
  Coincidentally, there are 1.609
  kilometers in a mile, which is within
  0.5% of the Golden ratio.
They're a much easier way to evaluate the function

$f(x) = \dfrac{\varphi^x -(-\varphi)^x}{\sqrt{5}}$

by hand, where $\varphi$ is the golden ratio and $x$ is an integer :)