Find nth Fibonacci number using Golden ratio

Last Updated : 02 Feb, 2023

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:
\varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots
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)                   f_n = round(f_n-1 * \varphi)

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 * \varphi) = 8 f7 = round(f6 * \varphi) = 13 f8 = round(f7 * \varphi) = 21 f9 = round(f8 * \varphi) = 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

// CPP program to find n-th Fibonacci number #include <bits/stdc++.h> using namespace std;  // Approximate value of golden ratio double PHI = 1.6180339;  // Fibonacci numbers upto n = 5 int f[6] = { 0, 1, 1, 2, 3, 5 };  // Function to find nth // Fibonacci number int 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 code int 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 ratio double PHI = 1.6180339;  // Fibonacci numbers upto n = 5 int f[6] = { 0, 1, 1, 2, 3, 5 };  // Function to find nth // Fibonacci number int 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 code int 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

// 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

# Python3 code to find n-th Fibonacci number  # Approximate value of golden ratio PHI = 1.6180339  # Fibonacci numbers upto n = 5 f = [ 0, 1, 1, 2, 3, 5 ]  # Function to find nth # Fibonacci number def 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 code n = 9 print(n, "th Fibonacci Number =", fib(n))  # This code is contributed by "Sharad_Bhardwaj".

// C# program to find n-th Fibonacci // number using 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 // PHP program to find n-th // Fibonacci number Approximate // value of golden ratio $PHI = 1.6180339;  // Fibonacci numbers  // upto n = 5  // Function to find nth // Fibonacci number function fib ($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 ?>

JavaScript

<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

Time complexity: O(n)
Auxiliary space: O(1)

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

Fast Doubling method to find the Nth Fibonacci number

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Find nth Fibonacci number using Golden ratio

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