Binet's Formula Calculator

Formula

The formula to calculate the Nth Fibonacci number is:

\[ F_n = \frac{\phi^n - \psi^n}{\sqrt{5}} \]

Where:

\(F_n\) is the Nth Fibonacci number
\(\phi\) (phi) is the golden ratio, approximately 1.61803398875
\(\psi\) (psi) is the conjugate of the golden ratio, approximately -0.61803398875
\(n\) is the Nth term in the Fibonacci sequence

What is Binet's Formula?

Binet's formula is an explicit formula used to find the Nth term of the Fibonacci sequence without having to calculate all the preceding terms. It is derived from the closed-form solution of the Fibonacci sequence and involves the golden ratio, \(\phi\), and its conjugate, \(\psi\). This formula allows for the rapid calculation of large Fibonacci numbers.

Example Calculation

Let's assume the following value:

Nth Term (n) = 10

Using the formula:

\[ F_{10} = \frac{1.61803398875^{10} - (-0.61803398875)^{10}}{\sqrt{5}} \approx 55 \]

The 10th Fibonacci number is approximately 55.