# Python program to find the nth Fibonacci Number

In the following tutorial, we will understand how to find the nth Fibonacci Number using Python. We can define a Fibonacci Number, where the following number is the sum of the preceding two numbers.

The first two elements of the Fibonacci series are 0 and 1, respectively. We can calculate the third element of the series by adding the preceding two elements and will get the third term as 0 + 1, which is equal to 1. Similarly, the fourth term will be the sum of the second and third terms, which is 1 + 1 = 2 and so on. The series of such numbers is known as a Fibonacci Series.

The recurrence relation defines a Fibonacci number as shown below:

Fn = Fn - 1 + Fn - 2

There are different ways to find the nth Fibonacci Number using the Python programming language. Some of them are as follows:

1. Finding nth Fibonacci Number using Recursion
2. Finding nth Fibonacci Number using dynamic programming
3. Finding nth Fibonacci Number using dynamic programming and space optimization
4. Finding nth Fibonacci Number using arrays

Of these ways, the two most fundamental are the Recursion method and the Dynamic method.

Let us understand the working of these methods in detail with examples.

## Finding nth Fibonacci Number using Recursion

The term recursion is used to define something within itself. In a programming language like Python, Recursion refers to the process of a function calling itself. With the proper and correct code, the Recursion will create a finite loop.

Let us consider the following snippet of code for better understanding.

Example:

Output:

```12th Element of the Fibonacci Series: 144
```

Explanation:

In the above snippet of code, we have defined a function as Fibonacci_Series() that accepts a parameter as n.

Moreover, we are aware that the first two elements of the Fibonacci are 0 and 1. In the event of the input as n = 1 or n = 2 (First or Second terms of Fibonacci series), we have used the if-else conditional statement to return 0 or 1. In case the value of n is greater than 2, the function will call itself with a lower input value. As we can observe that the code returns (Fibonacci_Series(n - 1) + Fibonacci_Series(n - 2)). Here, the function calls itself with a lower value unless it reaches the base value of n = 1 and n = 2, and as we know from before, n = 1 returns 0 and n = 2 returns 1. The returned values are then continuously added to produce the sequence of the Fibonacci Series.

## Finding the nth Fibonacci Number using Dynamic Programming

Dynamic Programming utilizes Recursion as well; however, it mainly utilizes if-else conditional statements. Within the statements, the value of the Fibonacci number is stored in a variable. With the help of Recursion, the repeated addition allows us to obtain this Fibonacci number.

Let us consider the following example to understand the same.

Example:

Output:

```12th Term of Fibonacci Series: 144
```

Explanation:

In the above snippet of code, we defined the function as Fibonacci_series(), which accepts the parameter as variable x. We created a one-dimensional array as fib_Array with data elements 0 and 1 in its zeroth and first indices. Then, if the provided input ('x') is less than or equal to 2, which is also the length of the array fib_Array, it returns 0 as the first number for x = 1 and 1 as the second number for x = 2. If the value of x is greater than 2, we have used recursion to call and insert the preceding two data elements. However, rather than returning the nth Fibonacci number directly, we append each of the summated elements to the fib_Array array. At last, we have returned the last element of the array (i.e., the nth element) and printed the value for the users.

## Finding the nth Fibonacci Number using Dynamic Programming and Space Optimization

This method is almost completely identical to Dynamic Programming. However, dynamic programming utilizes recursion to accomplish recurring addition, whereas this method utilizes the for-loop.

Let us consider the following example to understand the same.

Example:

Output:

```12th element of the Fibonacci Series: 144
```

Explanation:

In the above snippet of code, we have defined a function and assigned two variables, m = 0 and n = 1. These elements are the first and second elements of the Fibonacci Series. We have then used the if-elif-else conditional statements where the program returns 0 for input value x = 1 and 1 for input value x = 2. If the value of x is greater than 2, we have used the for-loop of i in the range (2, x + 1). We have taken a variable o to store the sum of the preceding two elements in the series. Once o takes the value of m + n, the value of m is reassigned to n. Subsequently, the value of n is reassigned to the value of o. This process continues, and value 3 keeps reassigning until the loop terminates. Once the loop is terminated, the function returns the value of n, which stores the value of the nth Fibonacci Number.

## Finding the nth Fibonacci Number using Array

In this method, we create an array of size x by repeated addition using the for-loop. Hence, the nth Fibonacci Number is returned.

Let us consider the following example to understand the same.

Example:

Output:

```12th element of the Fibonacci series: 144
```

Explanation:

In the above snippet of code, we have defined the function. Within the function, we have created an array to find the nth element of the Fibonacci Series. We have then used the for-loop to add elements of the series to the array by repeating the addition of the preceding two elements. At last, the nth element is returned and printed for the users.

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