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 an 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.