Binet's Formula

The Fibonacci series is defined recursively. That is, in order to find each term of the series using the definition, you have to find all the terms that precede it. This makes finding the nth term very difficult for large values of n, as you must find every term that comes before.

However, there could be a way to find Fibonacci numbers without using the definition. If this were possible, one would be able to find the nth term of the series simply by plugging n into a mathematical formula.

In 1843, Jacques Philippe Marie Binet discovered just such a formula for finding the nth term of the Fibonacci series. The formula itself looks like this:

The proof of this equation is dependant on concepts we have not yet explored, and will be given in full in a later lesson.

Binet's formula involves two very special numbers, the two numerical expressions within the lowest sets of parentheses. Remember these two numbers: they will become very important in later sections.