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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.

using System;

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

  static void Main(string[] args) {
    Console.WriteLine("Fibonacci 5th term: " + fib(5));
    Console.WriteLine("Fibonacci 6th term: " + fib(6));
    Console.WriteLine("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.

using System;

class MyProgram {
  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];  
  }  

  static void Main(string[] args) {
    Console.WriteLine("Fibonacci 6th term: " + fib(6));
    Console.WriteLine("Fibonacci 7th term: " + fib(7));
    Console.WriteLine("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.

using System;

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

  static void Main(string[] args) {
    Console.WriteLine("Fibonacci 8th term: " + fib(8));
    Console.WriteLine("Fibonacci 9th term: " + fib(9));
    Console.WriteLine("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.

using System;

class MyProgram {
  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;  
  }  

  static void Main(string[] args) {
    Console.WriteLine("Fibonacci 9th term: " + fib(9));
    Console.WriteLine("Fibonacci 10th term: " + fib(10));    
    Console.WriteLine("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.

using System;
 
class MyProgram {
  static int fib(int n) {
    int[,] initial = new int[2,2]{{1,1},{1,0}};
    int[,] Final = new int[2,2]{{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];  
  }  

  static void Main(string[] args) {
    Console.WriteLine("Fibonacci 10th term: " + fib(10));    
    Console.WriteLine("Fibonacci 11th term: " + fib(11));
    Console.WriteLine("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




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