Binet’s Formula Calculator

To find the $n$ -th Fibonacci number using Binet’s Formula, substitute $n$ into the formula and solve using the values of φ and ψ.

The Binet’s Formula Calculator is a tool that is developed to find any Fibonacci number without needing to go through all previous numbers in the sequence. Moreover, Binet’s formula provides a mathematical shortcut to obtain the $n$ -th term directly.

This formula uses the golden ratio, denoted as φ, and its negative reciprocal, denoted as ψ. It is also worth-noting that it’s named after the French mathematician Jacques Philippe Marie Binet.

$n$ Formula :

 $F_{n} = \frac{\phi^{n} - \psi^{n}}{\sqrt{5}}$

Variable	Description
$F_{n}$	$n$ -th Fibonacci number
$\phi$	Golden ratio, approximately 1.618
$\psi$	Negative reciprocal of $\phi$ , ~ -0.618
$n$	Position of the term in the Fibonacci sequence

How to Use Binet’s Formula:

Identify the nth term you want to find.
Apply the values of $\phi$ and the conjugate of $\phi$ .
Plug these into the formula and simplify.

Solved Calculation:

Example 1:

Given: $n = 5$

Step	Calculation
Substitute values	$F_{5} = \frac{1.618^{5} – (-0.618)^{5}}{\sqrt{5}}$
Result	$F_{5} = \frac{11.09 – (-0.09)}{2.236}$
Final answer	$F_{5} = 5$

Answer: The 5th Fibonacci number is 5.

Example 2:

Given: $n = 8$

Step	Calculation
Substitute values	$F_{8} = \frac{1.618^{8} – (-0.618)^{8}}{\sqrt{5}}$
Result	$F_{8} = \frac{76.03 – 0.03}{2.236}$
Final answer	$F_{8} = 21$

Answer: The 8th Fibonacci number is 21.

What is Binet’s Formula Calculator?

Binet’s Formula Calculator is a classic tool that is used to find the nth term of the Fibonacci sequence. As a matter of fact, it applies Binet’s Formula. This formula provides a direct way to calculate the Fibonacci number at a specific position without having to generate all previous numbers.

Where it is used ?

Mathematical sequences: To find Fibonacci terms without iterative calculations.
Golden Ratio applications: To explore patterns related to nature, architecture, and design.
Computer algorithms: To solve recursive problems efficiently.

More advanced Tools :

Online calculators that include step-by-step solutions.
Fibonacci calculators to explore relationships with the Golden Ratio.
Fibonacci sequence nth-term calculators for quickly finding any term in the series

Final Words:

All in all, this formula makes it possible to calculate any Fibonacci number directly, making it a powerful mathematical tool.

Contents

1 Binet’s Formula Calculator