Find nth Fibonacci number using Golden ratio

24 Jun, 2022
Fibonacci series = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ........Different methods to find nth Fibonacci number are already discussed. Another simple way of finding nth Fibonacci number is using golden ratio as Fibonacci numbers maintain approximate golden ratio till infinite. Golden ratio: Examples:  
Input : n = 9
Output : 34

Input : n = 7
Output : 13
 
Approach: Golden ratio may give us incorrect answer. We can get correct result if we round up the result at each point.  
nth fibonacci number = round(n-1th Fibonacci number X golden ratio)
                  fn = round(fn-1 * )
Till 4th term, the ratio is not much close to golden ratio (as 3/2 = 1.5, 2/1 = 2, ...). So, we will consider from 5th term to get next fibonacci number. To find out the 9th fibonacci number f9 (n = 9) :  
     f6 = round(f5 * ) = 8f7 = round(f6 * ) = 13f8 = round(f7 * ) = 21f9 = round(f8 * ) = 34
Note: This method can calculate first 34 fibonacci numbers correctly. After that there may be difference from the correct value. Below is the implementation of above approach:  
CPP
C
Java
Python3
C#
PHP
Javascript
// CPP program to find n-th Fibonacci number#include <bits/stdc++.h>using namespace std; // Approximate value of golden ratiodouble PHI = 1.6180339; // Fibonacci numbers upto n = 5int f[6] = { 0, 1, 1, 2, 3, 5 }; // Function to find nth// Fibonacci numberint fib(int n){    // Fibonacci numbers for n < 6    if (n < 6)        return f[n];     // Else start counting from    // 5th term    int t = 5, fn = 5;    while (t < n) {        fn = round(fn * PHI);        t++;    }    return fn;} // driver codeint main(){    int n = 9;    cout << n << "th Fibonacci Number = " << fib(n) << endl;    return 0;} // This code is contributed by Sania Kumari Gupta// (kriSania804)
// C program to find n-th Fibonacci number#include <math.h>#include <stdio.h> // Approximate value of golden ratiodouble PHI = 1.6180339; // Fibonacci numbers upto n = 5int f[6] = { 0, 1, 1, 2, 3, 5 }; // Function to find nth// Fibonacci numberint fib(int n){    // Fibonacci numbers for n < 6    if (n < 6)        return f[n];     // Else start counting from    // 5th term    int t = 5, fn = 5;     while (t < n) {        fn = round(fn * PHI);        t++;    }     return fn;} // driver codeint main(){    int n = 9;    printf("%d th Fibonacci Number = %d\n", n, fib(n));    return 0;} // This code is contributed by Sania Kumari Gupta// (kriSania804)
// Java program to find n-th Fibonacci number class GFG{    // Approximate value of golden ratio    static double PHI = 1.6180339;         // Fibonacci numbers upto n = 5    static int f[] = { 0, 1, 1, 2, 3, 5 };         // Function to find nth    // Fibonacci number    static int fib (int n)    {        // Fibonacci numbers for n < 6        if (n < 6)            return f[n];             // Else start counting from        // 5th term        int t = 5;        int fn = 5;             while (t < n) {            fn = (int)Math.round(fn * PHI);            t++;        }             return fn;    }         // Driver code    public static void main (String[] args)    {        int n = 9;        System.out.println(n + "th Fibonacci Number = "                                                +fib(n));    }} // This code is contributed by Anant Agarwal.
# Python3 code to find n-th Fibonacci number # Approximate value of golden ratioPHI = 1.6180339 # Fibonacci numbers upto n = 5f = [ 0, 1, 1, 2, 3, 5 ] # Function to find nth# Fibonacci numberdef fib ( n ):     # Fibonacci numbers for n < 6    if n < 6:        return f[n]     # Else start counting from    # 5th term    t = 5    fn = 5         while t < n:        fn = round(fn * PHI)        t+=1         return fn # driver coden = 9print(n, "th Fibonacci Number =", fib(n)) # This code is contributed by "Sharad_Bhardwaj".
// C# program to find n-th Fibonacci// numberusing System; class GFG {         // Approximate value of golden ratio    static double PHI = 1.6180339;         // Fibonacci numbers upto n = 5    static int []f = { 0, 1, 1, 2, 3, 5 };         // Function to find nth    // Fibonacci number    static int fib (int n)    {                 // Fibonacci numbers for n < 6        if (n < 6)            return f[n];             // Else start counting from        // 5th term        int t = 5;        int fn = 5;             while (t < n) {            fn = (int)Math.Round(fn * PHI);            t++;        }             return fn;    }         // Driver code    public static void Main ()    {        int n = 9;                 Console.WriteLine(n + "th Fibonacci"                    + " Number = " + fib(n));    }} // This code is contributed by vt_m.
<?php// PHP program to find n-th// Fibonacci number Approximate// value of golden ratio$PHI = 1.6180339; // Fibonacci numbers// upto n = 5 // Function to find nth// Fibonacci numberfunction fib ($n){    global $PHI;    $f = array(0, 1, 1, 2, 3, 5);         // Fibonacci numbers    // for n < 6    if ($n < 6)        return $f[$n];     // Else start counting    // from 5th term    $t = 5;    $fn = 5;     while ($t < $n)    {        $fn = round($fn * $PHI);        $t++;    }     return $fn;}     // Driver Code    $n = 9;    echo $n, "th Fibonacci Number = ",        fib($n), "\n"; // This code is contributed by aj_36?>
<script> // JavaScript program to find n-th Fibonacci number   // Approximate value of golden ratio    let PHI = 1.6180339;       // Fibonacci numbers upto n = 5    let f = [ 0, 1, 1, 2, 3, 5 ];       // Function to find nth    // Fibonacci number    function fib (n)    {        // Fibonacci numbers for n < 6        if (n < 6)            return f[n];           // Else start counting from         // 5th term        let t = 5, fn = 5;           while (t < n) {             fn = Math.round(fn * PHI);             t++;        }           return fn;       }       // driver code     let n = 9;     document.write(n + "th Fibonacci Number = " + fib(n) + "<br>");   // This code is contributed by Mayank Tyagi </script>
Output:  
9th Fibonacci Number = 34
We can optimize above solution work in O(Log n) by using efficient method to compute power.The above method may not always produce correct results as floating point computations are involved. This is the reason, this method is not used practically even if it can be optimized to work in O(Log n). Please refer below MIT video for more details.https://www.youtube.com/watch?v=-EQTVuAhSFY 
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