# Java Program - Fibonacci Sequence

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 example below, 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:

To calculate Fn,is 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
```

5