Binomial Form

There is one more way that we know to calculate the nth term of the Fibonacci series; a very complicated one, in fact. This method is recommended to those already familiar with summation notation, binomial theory, and Pascal's Triangle.

Every Fibonacci number can be expressed as the sum of a certain number of entries in Pascal's Triangle, as such:

Note that the notation stands for "binomial n, k" and is the entry in the nth row and kth column of Pascal's Triangle. However, when finding rows and columns of Pascal's Triangle, one must be very careful to remember that the first row and column is always counted as 0. Therefore, the first row of Pascal's Triangle is actually the second one down, etc.

Also, it is important to notice one other thing about binomial notation. If the lower number between the parentheses is larger than the upper number, the value of the term is always zero. That is, if k is greater than n, then "binomial n, k" is zero.

This equation can seem very complex, especially to those not totally familiar with the concepts it involves. For those looking for a more visual example of how the equation works, see the "In Action" page for this section.

Random Fibonacci number from the 1st through the 200th:

All contents, unless otherwise specified, are © 1999 by Matt Anderson, Jeffrey Frazier, and Kris Popendorf.
Created by Team 27890 for the 1999 competition.