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Fibonacci terms are generally represented as Fn. A Fibonacci term is the sum of two previous terms and starts with 0 and 1. Mathematically, it can be represented as:

Fn = Fn-1 + Fn-2

With boundary conditions: F0 = 0 and F1 = 1

The Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233...

Method 1: Using Recursive method

In the below example, a recursive method called fib() is created to find out the nth term of Fibonacci sequence.

public class MyClass {
  static int fib(int n) {
    if (n == 0)
      return 0;
    else if (n == 1)
      return 1;
    else
      return fib(n-1) + fib(n-2);
  }

  public static void main(String[] args) {
    System.out.println("Fibonacci 5th term: " + fib(5));
    System.out.println("Fibonacci 6th term: " + fib(6));
    System.out.println("Fibonacci 7th term: " + fib(7));
  }
}

The above code will give the following output:

Fibonacci 5th term: 5
Fibonacci 6th term: 8
Fibonacci 7th term: 13

Method 2: Using Dynamic Programming

The Fibonacci term can also be estimated using dynamic programming. As compared to the recursive method, it calculates a specific term of the sequence only once.

public class MyClass {
  static int fib(int n) {
    //creating array which contains Fibonacci terms
    int[] f = new int[n+1];
    f[0] = 0;
    f[1] = 1;
    for(int i = 2; i <= n ; i++) {
      f[i] = f[i-1] + f[i-2];
    }
    return f[n];
  }

  public static void main(String[] args) {
    System.out.println("Fibonacci 6th term: " + fib(6));
    System.out.println("Fibonacci 7th term: " + fib(7));
    System.out.println("Fibonacci 8th term: " + fib(8));
  }
}

The above code will give the following output:

Fibonacci 6th term: 8
Fibonacci 7th term: 13
Fibonacci 8th term: 21

Method 3: Using Ternary Operator

This can also be achieved using ternary operator.

public class MyClass {
  static int fib(int n) {
    int y = (n == 0)? 0 : (n == 1) ? 1 : fib(n-1) + fib(n-2);
    return y;
  }

  public static void main(String[] args) {
    System.out.println("Fibonacci 8th term: " + fib(8));
    System.out.println("Fibonacci 9th term: " + fib(9));
    System.out.println("Fibonacci 10th term: " + fib(10));    
  }
}

The above code will give the following output:

Fibonacci 8th term: 21
Fibonacci 9th term: 34
Fibonacci 10th term: 55

Method 4: Space optimized method

In this method, only three variables are used which changes in each iteration and finally nth term of Fibonacci Sequence is calculated.

public class MyClass {
  static int fib(int n) {
    int a = 0, b = 1, c = 0;

    if (n == 0)
      return a;

    for(int i = 2; i <= n; i++) {
      c = a + b;
      a = b;
      b = c;    
    }
    
    return b;
  }

  public static void main(String[] args) {
    System.out.println("Fibonacci 9th term: " + fib(9));
    System.out.println("Fibonacci 10th term: " + fib(10));    
    System.out.println("Fibonacci 11th term: " + fib(11));
  }
}

The above code will give the following output:

Fibonacci 9th term: 34
Fibonacci 10th term: 55
Fibonacci 11th term: 89

Method 5: Using power of matrix

A Fibonacci sequence term can also be calculated as power of matrix. A Fibonacci sequence holds below mentioned property:

power of matrix

To calculate Fn,power of matrixis calculated and A01 will be the Fn.

public class MyClass {
  static int fib(int n) {
    int[][] initial = {{1,1},{1,0}};
    int[][] Final = {{1,1},{1,0}};
    int a, b, c, d;
    if (n == 0)
      return 0;
    else {      
      for(int i = 1; i < n ; i++){
        a = Final[0][0]*initial[0][0] + Final[0][1]*initial[1][0];
        b = Final[1][0]*initial[0][0] + Final[1][1]*initial[1][0];
        c = Final[0][0]*initial[0][1] + Final[0][1]*initial[1][1];
        d = Final[1][0]*initial[0][1] + Final[1][1]*initial[1][1];
        Final[0][0] = a;
        Final[1][0] = b;
        Final[0][1] = c;
        Final[1][1] = d;           
      }
    }
    return Final[0][1];
  }

  public static void main(String[] args) {
    System.out.println("Fibonacci 10th term: " + fib(10));    
    System.out.println("Fibonacci 11th term: " + fib(11));
    System.out.println("Fibonacci 12th term: " + fib(12));
  }
}

The above code will give the following output:

Fibonacci 10th term: 55
Fibonacci 11th term: 89
Fibonacci 12th term: 144