Phi and the Fibonacci Series

Phi and the Fibonacci Series

Leonardo Fibonacci discovered the series which converges on phi

In the 12th century, Leonardo Fibonacci discovered a simple numerical series that is the foundation for an incredible mathematical relationship behind phi.
Starting with 0 and 1, each new number in the series is simply the sum of the two before it.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 . . .

The ratio of each successive pair of numbers in the series approximates phi (1.618. . .) , as 5 divided by 3 is 1.666..., and 8 divided by 5 is 1.60.

The table below shows how the ratios of the successive numbers in the Fibonacci series quickly converge on Phi. After the 40th number in the series, the ratio is accurate to 15 decimal places.
1.618033988749895 . . .

Compute any number in the Fibonacci Series easily!

Here are two ways you can use phi to compute the nth number in the Fibonacci series (fn).
If you consider 0 in the Fibonacci series to correspond to n = 0, use this formula:
fn = Phi n / 5½
Perhaps a better way is to consider 0 in the Fibonacci series to correspond to the 1st Fibonacci number where n = 1 for 0. Then you can use this formula, discovered and contributed by Jordan Malachi Dant in April 2005:
fn = Phi n / (Phi + 2)
Both approaches represent limits which always round to the correct Fibonacci number and approach the actual Fibonacci number as n increases.