Find Nth Fibonacci Number using Binet’s Formula

Given a number n, print n-th Fibonacci Number, using Binet’s Formula.

Examples:

Input: n = 5

Output: 1

Input: n = 9

Output: 34

What is Binet’s Formula?

Binet’s Formula states that:

If  is the  Fibonacci number, then

Why is Binet’s Formula not used regularly?

Binet’s Formula gives correct results only up to n<71.

Because as we move forward from n>=71, the rounding error becomes significantly large. Although, using the floor function instead of the round function will give the correct result for n=71. But after n=72, it also fails.

Example: For N=72 , Correct result is 498454011879264 but above formula gives 498454011879265.

Approach to Find Nth Fibonacci Number using Binet’s Formula

In this method, we directly implement the formula for the nth term in the Fibonacci series. F_n = {[(√5 + 1)/2] ^ n} / √5 

Below is the implementation of the above approach:

C++

// C++ Program to find n'th fibonacci Number
#include <cmath>
#include <iostream>

int fib(int n)
{
    double phi = (1 + sqrt(5)) / 2;
    return round(pow(phi, n) / sqrt(5));
}

// Driver Code
int main()
{
    int n = 9;
    std::cout << fib(n) << std::endl;
    return 0;
}
// This code is contributed by Lokesh Mohanty.

Java

// Java Program to find n'th fibonacci Number
import java.util.*;

public class GFG {

    static int fib(int n)
    {
        double phi = (1 + Math.sqrt(5)) / 2;
        return (int)Math.round(Math.pow(phi, n)
                               / Math.sqrt(5));
    }

    // Driver Code
    public static void main(String[] args)
    {
        int n = 9;
        System.out.println(fib(n));
    }
};
// This code is contributed by PrinciRaj1992

Python3

# Python3 program to find n'th
# fibonacci Number
import math


def fibo(n):
    phi = (1 + math.sqrt(5)) / 2

    return round(pow(phi, n) / math.sqrt(5))


# Driver code
if __name__ == '__main__':

    n = 9

    print(fibo(n))

# This code is contributed by prasun_parate

C#

// C# Program to find n'th fibonacci Number
using System;

public class GFG {
    static int fib(int n)
    {
        double phi = (1 + Math.Sqrt(5)) / 2;
        return (int)Math.Round(Math.Pow(phi, n)
                               / Math.Sqrt(5));
    }

    // Driver code
    public static void Main()
    {
        int n = 9;
        Console.WriteLine(fib(n));
    }
}

// This code is contributed by 29AjayKumar

Javascript

// Javascript Program to find n'th fibonacci Number
function fib(n) {
  let phi = (1 + Math.sqrt(5)) / 2;
  return Math.round(Math.pow(phi, n) / Math.sqrt(5));
}

let n = 9;
console.log(fib(n));

// This code is contributed by mukesh07.

PHP

<?php
// PHP Program to find n'th
// fibonacci Number

function fib($n)
{
    $phi = (1 + sqrt(5)) / 2;
    return round(pow($phi, $n) / sqrt(5));
}

// Driver Code
$n = 9;
echo fib($n) ;

// This code is contributed by Ryuga
?>

Output

34

Time Complexity: O(log n), this is because calculating phi^n takes log n time
Auxiliary Space: O(1)