Write a recursive and an iterative function to calculate the nth element in a Fibonacci sequence. A Fibonacci sequence is defined as the element 1, followed by another 1, and each element thereafter is the sum of the previous two elements. For example, the first 9 elements of a Fibonacci sequence are: 1 2 3 5 8 13 21 34 This famous sequence was originally used to predict the growth of rabbit populations! Once you have each of the functions working for n equal to 40, determine which method is more efficient by timing the two separate function calls and printing out the time required for each method call to return the 40th element in the sequence. Return the 40th element to main and print it. After that, print out a complete Fibonacci sequence from element 1 to element 40, along with its position number. 1 1 2 3

Here is the program that contains a recursive and an iterative function to calculate the nth element in a Fibonacci sequence :

import java.util.Scanner;   // to take input from user

public class Main {  

   public static void main(String[] args) {  //start of main function

  Scanner input = new Scanner(System.in);   // creates Scanner class object

       System.out.print("Enter the value of n: ");  //prompts user to enter n

       int n = input.nextInt();   // reads the input value of n from user

       System.out.println("Fibonacci using iteration:");  // displays result of iterative method

       long start = System.currentTimeMillis();  //computes the time required by iterative method

       System.out.printf("Element at index %d = %d \n", n, Iteration(n));   //returns the nth element of fibonacci series      

 System.out.printf("Time taken: %d ms\n", System.currentTimeMillis() - start);  //displays the time taken to return the nth element of fibonacci series using iterative method

        int  i = 0, j;

      for (  j = 1 ; j <= n ; j++ )    {  //used to print the fibonacci sequence using iterative method

     System.out.printf("%d : %d\n",j, Iteration(i));

     i++;     }  

       System.out.println("Fibonacci using recursion:");  // displays result of recursive method

       start = System.currentTimeMillis();  //computes the time required by recursive method

       System.out.printf("Element at index %d = %d \n", n, Recursion(n));  //returns the nth element of fibonacci series  

       System.out.printf("Time: %d ms\n", System.currentTimeMillis() - start);  //displays the time taken to return the nth element of fibonacci series using recursive method

          int  ir = 0, jr;

      for (  jr = 1 ; jr <= n ; jr++ )    {

     System.out.printf("%d : %d\n",jr, Iteration(ir));  //used to print the fibonacci sequence using recursive method

     ir++;     }     }  

   static int Iteration(int n) {  // iterative method for fibonacci series

       int a = 0, b = 1, c = 1;  

       for (int i = 0; i < n; i++) {  

           a = b;  

           b = c;

           c = a + b;         }  

       return a;     }  

   static int Recursion(int  n) {  // recursive method for fibonacci series

       if ((n == 1) || (n == 0)) {  //base case

           return n;         }

       return Recursion(n - 1) + Recursion(n - 2);     } } //recursive case

Explanation:

The explanation is provided in the attached document.