Fibonacci Series In Python & Nth Term | Generate & Print (+Codes)

The Fibonacci series is one of the most well-known sequences in mathematics. Named after the Italian mathematician Leonardo of Pisa, known as Fibonacci, this series appears in various natural phenomena. It also has numerous applications in computer science, financial markets, and algorithm development. In this article, we will explore what this series is and how to generate the Fibonacci series in Python programming using various techniques.

What Is The Fibonacci Series?

The Fibonacci series is a sequence of numbers in which each number is the sum of the two preceding ones, typically starting with 0 and 1(seed values). The series appears in many different areas of mathematics and science, often in contexts with growth patterns and natural formations. In mathematical terms, the Fibonacci series is defined by the following recurrence relation:

F(n)=F(n−1)+F(n−2)
with the initial conditions:
F(0)=0 (0th term)
F(1)=1 (1st term)

This means that to find any number in the Fibonacci series, you add the two previous numbers, i.e., the (n-1)th term and the (n-2)th term in the series. The series usually starts as 0,1,1,2,3,5,8,13,21,34,…

Practical Occurrences of the Fibonacci Series:

The Fibonacci sequence is not just a theoretical construct; it appears in various natural and practical contexts:

• Botanical Patterns: Many plants exhibit Fibonacci numbers in the arrangement of leaves, seeds, and flowers. For example, the number of petals on a flower is often a Fibonacci number. Lilies have 3 petals, buttercups have 5, and daisies can have 34, 55, or even 89 petals.

• Rabbit Population Growth: Fibonacci originally introduced the series in the context of a problem involving the growth of a rabbit population. Suppose a pair of rabbits can reproduce starting at one month old and continue to produce another pair every subsequent month. The population growth of rabbits can be modelled by the Fibonacci series: 0 pairs at month 0, 1 pair at month 1, 1 pair at month 2, 2 pairs at month 3, 3 pairs at month 4, 5 pairs at month 5, and so on.

Visual Representation Of The Fibonacci Series

One of the most visually appealing representations of the Fibonacci series is the Fibonacci spiral. This is created by drawing quarter-circle arcs inside squares with side lengths that are Fibonacci numbers. When these squares are placed adjacently, a spiral pattern emerges, which can often be seen in natural formations such as shells and hurricanes.

Pseudocode Code For Fibonacci Series Program In Python

Here's a pseudocode to compute the Fibonacci series in Python language:

# Function to compute the Fibonacci series
function fibonacci(n):
    if n <= 1:
        return n
    else:
        # Initialize variables for the first two Fibonacci numbers
        a = 0
        b = 1
    # Iterate from 2 to n to compute subsequent Fibonacci numbers
    for i from 2 to n:
        # Compute the next Fibonacci number by summing the previous two numbers
        c = a + b
        # Update variables for the next iteration
        a = b
        b = c
        return b

# Example
n = 10
print("Fibonacci series up to", n, "terms:")
for i from 0 to n-1:
    print(fibonacci(i), end=" ")

The pseudocode for the Fibonacci series in Python outlines a function to compute the Fibonacci numbers up to a given limit n.

• It initializes variables for the first two Fibonacci numbers, a and b, and then iteratively computes subsequent Fibonacci numbers using a loop.
• Each Fibonacci number is the sum of the two preceding numbers. Finally, it returns the last computed Fibonacci number.

Generating Fibonacci Series In Python Using Naive Approach (While Loop)

The native approach to generating the Fibonacci series involves using a while loop to iteratively compute each term of the series. The while loop iteratively executes a block of code till the time the loop condition is true. This simplest method is straightforward and easy to understand, making it a good starting point for beginners.

Below is a simple Python program example that illustrates the use of this approach to generate a Fibonacci series and print the n-th term.

Code Example:

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Output:

Fibonacci series up to 6 terms: [0, 1, 1, 2, 3, 5]
The 6-th Fibonacci term: 8

Explanation:

In the simple Python code example-

1. We define the fibonacci_series_and_nth_term(n) function to generate the Fibonacci series up to n terms and compute the n-th Fibonacci number using a naive approach with a while loop.
2. Inside the function, we first have a docstring to describe it. Then, we enter an if-elif statement where the if-condition checks whether n is less than equal to 0. 
   • If the condition n<=0 is true, then the if-block is executed, and an empty list is returned and 0.
   • If the condition n<=0 is false, we move to the elif-condition to check if n is equal to 1, i.e., n==1.
   • If this condition is true, elif-block is executed, returning a list containing only 0 and the number 0 as the nth Fibonacci number.
3. If the condition n==0 is also false, we move to the next line of the function and initialize a list called series with 2 values, 0 and 1.
4. Then we initialize variables a and b to 0 and 1, respectively, variable nth_term to 0, and a variable count to 2. At this point, we already have two terms in the series.
5. We then enter a while loop to iteratively compute the next terms of the Fibonacci series. The loop condition checks if the variable count is less than n.
   • If the loop condition is met, we calculate the value of the next_term variable (which represents the net term in the series) by adding variables a to b.
   • After that, we use the append() function to next_term to the series list.
   • Then, we update the values of variables a to b, b to the next term, and increment count by 1 before the next iteration.
6. After the loop completes iterations, the list series will contain the Fibonacci series and we compute the n-th Fibonacci number.
7. Then, we compute the nth_term variable, which represents the nth number of the series using a formula with an if-else condition. Here, if n is greater than 2, we calculate it as the sum of the last two terms in the series. Otherwise, it's 1.
8. Finally, the function returns the series list and the n-th Fibonacci number.
9. In the main part of the program, we initialize a variable n with the value 6. This represents the number of elements in the series.
10. We then call the fibonacci_series_and_nth_term() function, passing n as an argument.
11. The function calculates the Fibonacci series up to 6 terms, and the 6-th Fibonacci term and outcome are stored in variables fibonacci_series and nth_term, respectively.
12. Lastly, we use a set of print() statements to display these values to the console.

Time Complexity: O(n)
Space Complexity: O(1)

Fibonacci Series Program In Python Using The Direct Formula

Binet's formula provides a direct mathematical expression to compute the n-th Fibonacci number without iterating through the series. We can leverage this formula to generate the nth terms of the Fibonacci series in Python. This approach offers a concise and efficient solution, particularly for large values of n, bypassing the need for iterative or recursive computation.

The Binet formula for finding the n-th Fibonacci number is as follows:

This direct formula computes the n-th Fibonacci number directly without requiring intermediate computations or storage, making it highly efficient and suitable for situations where high-performance calculation is essential. Below is a sample Python program that illustrates the use of this formula to generate the number of the Fibonacci series.

Code Example:

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Output:

The 7-th Fibonacci number: 13

Explanation:

In the sample Python code, we first import the math module.

1. Then, we define the fibonacci_direct_formula(n) function to compute the n-th Fibonacci number using the direct formula, i.e., the Binet Formula.
2. Inside the function, we first define a variable phi, i.e., the golden ratio as (1 + math.sqrt(5)) / 2. We use the sqrt() function from math module to calculate the square root of the 5.
3. Next, we apply Binet's formula for Fibonacci numbers, which states that the n-th Fibonacci number can be computed using the expression (phi ** n - (1 - phi) ** n) / sqrt(5).
4. The function returns the nearest integer to the value calculated by the formula, using the round() function and returns that as the n-th Fibonacci number.
5. In the main part, we initialize a variable n with the value 7 and then call the fibonacci_direct_formula() function, passing n as an argument.
6. The function calculates and returns the 7th term of the series, which is stored in the variable nth_term.
7. Lastly, we use a print() function to display this value to the console along with a string message using f-strings.

Time Complexity: O(1)
Space Complexity: O(n)

How To Generate Fibonacci Series In Python Using Recursion?

A recursive function is a special type of function that calls itself over and over again. We can write a Fibonacci series program in Python using recursion, where each Fibonacci number is computed by recursively calling the function itself with smaller input values until it reaches base cases. The recursive function definition method directly mirrors the mathematical definition of the Fibonacci sequence, making the code intuitive and easy to understand.

Look at the Python program sample below to understand how to define such a function to generate a Fibonacci series as well as the nth number.

Code Example:

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Output:

Fibonacci series up to 5 terms: [0, 1, 1, 2, 3]
The 5-th Fibonacci term: 5

Explanation:

In the Python code sample-

1. We define the function fibonacci_recursive(n), which takes an integer parameter n and computes the n-th Fibonacci number using recursion.
   • In this function, we define the base case using the if-else statement. The statement condition states that if n is less than or equal to 1, i.e., n<=1 is true, we return n directly.
   • If the condition is false, i.e., for values of n greater than 1, we recursively call fibonacci_recursive(n - 1) and fibonacci_recursive(n - 2) to compute the (n - 1)-th and (n - 2)-th Fibonacci numbers, respectively.
   • The function then returns the sum of these two values to obtain the n-th Fibonacci number.
2. Next, we define another function fibonacci_series_recursive(n), which also takes variable n as a parameter and generates the Fibonacci series up to n terms using recursion.
   • Inside this function, we use list comprehension to generate a list called series by calling fibonacci_recursive(i) for each index i from 0 to n - 1.
   • This will generate every Fibonacci number at the ith position till n and return the series.
3. In the main part, we initialize a variable n to the value 5 and call the fibonacci_series_recursive() function, passing n as an argument.
4. The invoked function generates the Fibonacci series up to 5 terms, and the outcome is stored in the variable fibonacci_series.
5. Next, we call the fibonacci_recursive() function by passing n as an argument. The function output here is the 5th Fibonacci number which is stored in the nth_term variable.
6. Lastly, we use a set of print() functions with f-strings to display both these values to the console. 

Time Complexity: O(n)
Space Complexity: O(n)

Generating Fibonacci Series In Python With Dynamic Programming

Dynamic programming is a method used to solve problems by breaking them down into simpler subproblems and storing the results of these subproblems to avoid redundant calculations. We can generate the Fibonacci series in Python using dynamic programming by iteratively building the sequence from the bottom up.

The dynamic programming approach for generating the Fibonacci series involves creating an array to store Fibonacci numbers as they are computed. This method avoids the exponential time complexity of the naive recursive approach by using an iterative process and maintaining a table of already computed Fibonacci values. The example Python program below illustrates the use of this approach to generate the Fibonacci series in Python and the nth term. 

Code Example:

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Output:

Fibonacci series up to 15 terms (Dynamic Programming): [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377]
The 15-th Fibonacci number (Dynamic Programming): 610

Explanation:

In the example Python code-

1. We define the function fibonacci_dynamic(n) to calculate the n-th Fibonacci number using dynamic programming.
2. Inside the function, we use an if-elif statement, which first checks if the value of n is less than or equal to 0. If the condition n<=0 is true, we return 0 as the base case.
   • If the elif condition n==1 is true, we return 1 as the second base case.
   • If none of the conditions are true, we move to the next line in the function.
3. Here, we create a list called fib of size n + 1 and initialize it with zeros.
4. Next, we initialize the element at the 1st position, i.e., fib[1] to 1, to establish the base case for the dynamic programming approach.
5. We then use a for loop to fill in the fib list from index 2 to n+1. For each index i, the element fib[i] is calculated by adding elements fib[i - 1] and fib[i - 2].
6. After the loop completes iterations, the function returns fib[n], which is the n-th Fibonacci number.
7. Next, we define another function, fibonacci_series_dynamic(n), to generate the Fibonacci series up to n terms using dynamic programming.
8. Inside this function, we again use an if-elif statement to check base cases n<=0 and n==1. If n<=0 is true, the function returns an empty list.
9. If the condition n==1 is true, then the function returns a list containing only 0. If none of the conditions are true, we move to the next line in the function.
10. Here, we create a list fib of size n, initialize all its elements with zeros and then set the element fib[1] to 1 to establish the base case for the dynamic programming approach.
11. Then, we use a for loop to fill in the fib list from index 2 to n - 1. For each index i, we calculate the element fib[i] by adding the elements fib[i - 1] and fib[i - 2].
12. After the loop terminates, the list fib contains the Fibonacci series which is returned by the function.
13. In the main section, we initialize a variable n_values to [15].
14. After that, we use a for loop to iterate over each nth element in n_values.
15. Inside the loop, we use a print() function to print the Fibonacci series up to n terms by calling the fibonacci_series_dynamic() function.
16. We then call the fibonacci_dynamic() function inside a print() statement to print the n-th Fibonacci number.

Time Complexity: O(n)
Space Complexity: O(n)

Fibonacci Series Program In Python Using For Loop

A for loop is generally used to iteratively execute a block of code for a certain range. We can use this loop construct to generate a Fibonacci series in Python. This approach includes initializing the first two terms of the series, typically 0 and 1. Then, iteratively compute each subsequent term by adding the last two terms.

• The process continues for a specified number of iterations, typically up to a given number of terms.
• Within the loop, the next term is calculated by summing the last two terms, and this term is appended to a list or printed directly.

This iterative approach efficiently produces the Fibonacci series in Python without the need for recursion, making it suitable for generating the series up to a desired number of terms. The Python program example below illustrates this approach.

Code Example:

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Output:

Fibonacci series up to 6 terms: [0, 1, 1, 2, 3, 5]

Explanation:

In the Python code example-

1. We define the function fibonacci_series_for_loop(n) to generate the Fibonacci series up to n terms using a for loop.
2. Inside the function, we first have an if-elif statement that checks conditions n<=0 and n==1. 
   • If the condition n<=0 is true, the function returns an empty list.
   • If the condition n==1 is true, the function returns a list containing only 0.
   • If none of the conditions are true, we move to the next line of the function.
3. Here, we initialize a list called series with the first two Fibonacci numbers: [0, 1].
4. Then, we use a for loop to iteratively compute the next terms of the Fibonacci series and add them to the list series. Inside the loop-
   • We begin from index 2, and in every iteration, calculate the next_term by taking the sum of the last two terms in the series.
   • Then, we use the append() function to add the element to the series list.
   • Once the loop finishes iterations, the list variable series will contain the Fibonacci series up to n terms.
5. In the main part of the program, we initialize a variable n with the value 6 and then call the fibonacci_series_for_loop() function by passing n as an argument.
6. The function invoked calculates and returns the Fibonacci series up to 6 terms and stores it in variable fibonacci_series.
7. Finally, we use a print() statement to display the Fibonacci series.

Time Complexity: O(n)
Space Complexity: O(1)

Generating Fibonacci Series In Python Using If-Else Statement

The if-else statement allows you to implement two alternative blocks of code depending on the condition being met. You can use this conditional statement to generate the Fibonacci series in Python.

• Here, you must first initialize the first two terms of the sequence, conventionally 0 and 1.
• Then, iteratively compute each subsequent term by adding the previous two terms.
• The if-else statement is utilized to handle special cases, like when the desired number of terms (n) is less than or equal to 0, 1, or 2, where the Fibonacci series follows specific patterns.
• For larger values of n, the series is generated within a loop where each term is calculated based on the last two terms. 

This approach offers a straightforward implementation, ensuring the correct sequence is generated while maintaining clarity in the code. The example program below illustrates how to generate and print the Fibonacci series in Python language. 

Code Example:

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Output:

Fibonacci series up to 7 terms: [0, 1, 1, 2, 3, 5, 8]

Explanation:

In the example code-

1. We define the function fibonacci_series_if_else(n) to generate the Fibonacci series up to n terms using if-else statements.
2. Inside the function, we first initialize an empty list called series to store the Fibonacci series.
3. Then, we have an if-statement, which checks if the value of n is less than or equal to 0. If the condition n<=0 is true, the function returns an empty list as there are no terms to generate.
4. We then append the first term, 0, to the series using the append() function.
5. The next if-statment checks whether the value of n equals 1. If the condition n==1 is true, the function returns the series containing only the first term.
6. Once again, we use the append() function to add the second term (i.e., 1) to the list series.
7. Another if-statement checks where the value of n equals 2. If the condition n==2 is true, the function returns the series containing the first two terms.
8. If none of the if-conditions are met, we move to a for loop that iteratively computes the terms of the Fibonacci series starting from the index position 2. 
   • The loop iterates till the loop control variable i becomes equal to n.
   • In every iteration, the next term of the series, represented by variable next_term, is calculated by adding the previous terms.
   • The next_term is then appended to the list series using the append() function.
   • Once the loop finishes iterations, the list series holds the Fibonacci series up to n terms.
9. In the main section, we initialize a variable n with the value 7 and call the fibonacci_series_if_else() function.
10. The function computes the Fibonacci series up to 7 terms and stores it in the variable fibonacci_series, which we print to the console using the print() function.

Time Complexity: O(n)
Space Complexity: O(1)

Note that the else part of the statement, while not written, is implicit in the structure of the Python code.

Generating Fibonacci Series In Python Using Arrays

When generating and printing a Fibonacci series in Python using arrays, you must first initialize an array to store the series. Then, iteratively compute each subsequent term using array operations.

• Typically, the first two terms of the series, 0 and 1, are initialized in the array.
• Subsequent terms are calculated by adding the last two terms in the array.
• This process continues until the desired number of terms is generated.
• The resulting array contains the Fibonacci series up to the specified length.
• Additionally, to compute the n-th Fibonacci number, the last element of the generated array is returned.

The example below illustrates this approach to provide you with a better understanding of the concept.

Code Example:

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

Output:

Fibonacci series up to 2 terms: [0, 1]
The 2-th Fibonacci term: 1

Explanation:

In the example-

1. We first define the function fibonacci_array(n) to generate the Fibonacci series up to n terms using arrays.
2. Inside the function, we have an if-elif-elif structure that checks conditions n<=0, n==1, and n==2.
   • If condition n<=0 is true, the function returns an empty list.
   • If condition n==1 is true, the function returns a list containing just [0].
   • If condition n==2 is true, the function returns a list containing [0, 1].
   • If none of the conditions are met, we move to the next line of the function.
3. Then, we initialize an array named series with the first two terms [0, 1].
4. Next, we use a for loop to iteratively compute the subsequent terms of the Fibonacci series from index 2 to n-1.
5. In each iteration, we compute the next term as the sum of the last two terms in the series (series[i - 1] + series[i - 2]) and append it to the series using the append() function.
6. After the loop completes execution, the array series contains the Fibonacci series up to n terms, which the function then returns.
7. Next, we define another function function, nth_fibonacci_array(n), to compute the n-th Fibonacci number using arrays.
   • Inside this function, we first call the fibonacci_array(n) to get the Fibonacci series up to n terms that is stored in the array series.
   • Then, we calculate and return the last element of the series as the n-th Fibonacci number if the series is not empty. Otherwise, we return None.  
8. To demonstrate the usage of these functions, we initialize a variable n with 2 in the main segment.
9. Then, we call the fibonacci_array(n) function, passing n as an argument. This function calculates the Fibonacci series and stores it in the variable fibonacci_series.
10. After that, we call the nth_fibonacci_array() function to compute the 2nd Fibonacci number directly and store it in the nth_term variable.
11. We print both the Fibonacci series and the 2nd Fibonacci number using print() function.

Time Complexity: O(n)
Space Complexity: O(n)

Generating Fibonacci Series In Python Using Cache

The caching approach to generating the Fibonacci series in Python optimizes computation by storing previously computed Fibonacci numbers in a cache, avoiding redundant calculations.

• It begins with initializing a cache to store Fibonacci numbers, typically starting with 0 and 1.
• Then, a recursive function computes Fibonacci numbers, utilizing the cache to retrieve previously computed values when available.

This method significantly improves efficiency, particularly for large Fibonacci numbers or series.

Code Example:

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

Output:

Fibonacci series up to 4 terms: [0, 1, 1, 2]
The 4-th Fibonacci term: 3

Explanation:

In the above code, we first import the lru_cache decorator from the functools module to enable caching (memoization).

1. We define the function fibonacci_cache(n) to compute the n-th Fibonacci number using a cache. We decorate this function with @lru_cache(maxsize=None) to enable unlimited caching.
2. Inside the function, an if-else statement checks if the value of n is less than or equal to 1. If the condition n<=1 is true, the function returns n (handling the base cases of 0 and 1).
3. For other values of n, the function recursively calls the fibonacci_cache(n - 1) and fibonacci_cache(n - 2) functions, adding their results to compute the n-th Fibonacci number.
4. Next, we define the function fibonacci_series_cache(n) to generate the Fibonacci series up to n terms using the cache.
5. Here, we use list comprehension to generate the series by calling fibonacci_cache(i) for each index from 0 to n - 1.
6. Then, the function returns the generated Fibonacci series, which is stored in the variable series.
7. To demonstrate the usage, we initialize the variable n with value 4 and call the fibonacci_series_cache() function in the main part.
8. The function computes the Fibonacci series up to 4 terms and stores it in the fibonacci_series variable.
9. Next, we call the fibonacci_cache() function to compute the 4-th Fibonacci number which it stores in the nth_term variable.
10. Finally, we print the Fibonacci series and the 4-th Fibonacci term using the print() function.

Time Complexity: O(n)
Space Complexity: O(n)

Generating Fibonacci Series In Python Using Backtracking

Generating the Fibonacci series in Python using backtracking involves recursively computing each Fibonacci number by exploring all possible paths in a tree-like structure until reaching the base cases.

This popular method employs a recursive function to compute each Fibonacci number, which recursively calls itself to compute the nth Fibonacci number by summing the n−1th and n−2th Fibonacci numbers. The basic Python program example below illustrates this approach.

Code Example:

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

Output:

Fibonacci series up to 5 terms: [0, 1, 1, 2, 3]
The 5-th Fibonacci term: 5

Explanation:

In the code-

1. We define the function fibonacci_backtracking() to compute the n-th Fibonacci number using a backtracking-like approach with memoization.
2. The function takes a variable n and a dictionary computed, which is initialized with base cases: 0 maps to 0, and 1 maps to 1, as parameters.
3. Inside the function, we have an if-else statement that checks if the n-th Fibonacci number is already computed by checking if n is in the computed dictionary. If the condition is true, the function returns the precomputed value.
4. If the value is not precomputed, we recursively compute the previous two Fibonacci numbers by calling fibonacci_backtracking(n - 1, computed) and fibonacci_backtracking(n - 2, computed) functions.
5. The outcomes of these functions are stored in the variables fib_n_minus_1 and fib_n_minus_1, respectively. 
6. Next, we store the sum of these two current values in the computed dictionary under the key n., which the function then returns as the computed n-th Fibonacci number.
7. After that, we define another function, fibonacci_series_backtracking(n), to generate the Fibonacci series up to n terms using the backtracking-like approach.
8. Inside the function, we declare a list called series and assign it a value using list comprehension to generate the series by calling fibonacci_backtracking(i) for each index from 0 to n - 1.
9. The function returns this list containing the Fibonacci series.
10. In the main part, we initialize a variable n with the value 5 and call the fibonacci_series_backtracking() function. 
11. This function computes the Fibonacci series up to 5 terms and stores it in fibonacci_series.
12. Following this, we call the fibonacci_backtracking() function to compute the 5th Fibonacci number stored in the nth_term variable.
13. Finally, we print the Fibonacci series and the 5th Fibonacci term using the print() function.

Time Complexity: O(2^n)
Space Complexity: O(2^n)

Fibonacci Series In Python Using Power Of Matix

The matrix exponentiation method for computing Fibonacci numbers leverages the properties of matrices to efficiently calculate the nth Fibonacci number. By representing the Fibonacci sequence as a matrix and using matrix exponentiation techniques, we can raise the Fibonacci matrix to the power of n to obtain the nth Fibonacci number directly.

Code Example:

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

Output:

Fibonacci series up to 7 terms: [0, 1, 1, 2, 3, 5, 8]
The 7-th Fibonacci term: 13

Explanation:

In the example, we begin by importing the NumPy module as np to use its functions.

1. We define the function fibonacci_matrix(n) to compute the n-th Fibonacci number using matrix exponentiation.
2. Inside the function, we use an if-else statement to handle the base cases with conditions n equals 0 and n equals 1.
   • If condition n==0 is true, the function returns 0.
   • If the condition n==1 is true, the function returns 1.
3. Then, we define the Fibonacci matrix called fib_matrix as a NumPy array, i.e., [[1, 1], [1, 0]], using the array() function from the numpy module.
4. Next, we raise the Fibonacci matrix to the power of (n - 1) using NumPy's np.linalg.matrix_power function, storing the resulting matrix in result_matrix.
5. We extract the Fibonacci number from the resulting matrix. Specifically, the element in the first row and first column (result_matrix[0][0]) represents the n-th Fibonacci number that the function returns.
6. After that, we define another function, fibonacci_series_matrix(n), to generate the Fibonacci series up to n terms using matrix exponentiation.
7. Inside, we use list comprehension to generate the series by calling the fibonacci_matrix function for each index from 0 to n - 1.
8. The outcome is stored in the list series, which the function returns.
9. To demonstrate the usage of these functions, we initialize the variable n with value 7 and call the fibonacci_series_matrix() function.
10. This invoked function computes the Fibonacci series up to 7 terms and stores it in fibonacci_series.
11. Next, we call the fibonacci_matrix() function with n to compute the 7-th Fibonacci number stored in the nth_term variable.
12. We print the Fibonacci series and the 7th Fibonacci term using the print() function.

Time Complexity: O( log n)
Space Complexity: O(log n)

Complexity Analysis For Fibonacci Series Programs In Python

The table below provides a comparison of the time and space complexities for each coding method to generate the Fibonacci series in Python, arranged in descending order.

The methods are arranged based on time complexity, with the most efficient methods to generate the Fibonacci series in Python listed at the top.

Applications Of Fibonacci Series In Python & Programming

The Fibonacci series finds applications in various fields due to its mathematical properties and recurrence relation. Some notable applications include:

1. Computer Science and Algorithms: Fibonacci numbers are used in algorithms related to dynamic programming, memoization, and optimization problems. They serve as a basis for developing efficient solutions to problems such as finding the shortest path in a graph, optimizing resource allocation, and generating permutations.
2. Mathematics: Fibonacci numbers appear in various mathematical concepts, such as number theory, combinatorics, and geometry. They are closely related to the golden ratio, which has applications in art, architecture, and design.
3. Finance and Economics: In finance, Fibonacci numbers and the golden ratio are applied in technical analysis to identify potential support and resistance levels in stock prices. Fibonacci retracement levels are commonly used by traders to predict market trends and make investment decisions. Additionally, Fibonacci sequences model population growth and the distribution of plants in botanical studies.
4. Natural Sciences: Fibonacci numbers are observed in natural phenomena, including the arrangement of leaves on a stem, the spirals of sunflower seeds, the branching patterns of trees, and the structure of pinecones and pineapples. These patterns arise due to the optimization of space and resources during growth processes, leading to the prevalence of Fibonacci sequences in nature.
5. Coding and Cryptography: Fibonacci numbers are used in coding theory and cryptography for error detection and correction, as well as in the generation of pseudorandom numbers. The mathematical properties of Fibonacci sequences contribute to the development of secure encryption algorithms and data transmission protocols.
6. Artificial Intelligence and Machine Learning: Fibonacci numbers are incorporated into machine learning algorithms for pattern recognition, sequence prediction, and feature extraction. They provide a mathematical framework for modelling and analyzing complex data structures and time series data.

Conclusion

There are many ways to generate the Fibonacci series in Python, including recursion, iteration, dynamic programming, and matrix exponentiation. Python provides versatile tools for computing Fibonacci numbers. The ability to generate and print the Fibonacci series in Python (as well as the nth Fibonacci number) can have significant applications across diverse fields, including computer science, finance, natural sciences, cryptography, and artificial intelligence.

Its patterns and properties not only inspire algorithmic innovations but also shed light on the underlying mathematical principles governing complex systems in nature and society. Python's flexibility and simplicity make it an ideal platform for experimenting with Fibonacci sequences. By harnessing the power of Python, programmers and researchers can delve deeper into the mysteries of the Fibonacci series and unlock its potential to solve problems.

Frequently Asked Questions

Q. What is the Fibonacci series in Python, and how is it defined?

The Fibonacci series in Python is a sequence of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. In mathematical terms, the Fibonacci series is defined recursively as:

F(0)=0
F(1)=1
F(n)=F(n−1)+F(n−2) for n≥2

In simpler terms, each Fibonacci number (except for the first two) is the sum of the two preceding Fibonacci numbers. This integer sequence starts with 0 and 1, and subsequent numbers are obtained by adding the previous two numbers. The Fibonacci series in Python is commonly used for various computational and mathematical purposes, such as algorithmic problem-solving, mathematical modelling, and generating sequences with interesting properties.

Q. What is the significance of the Fibonacci series in nature?

The Fibonacci series is observed in natural phenomena such as the arrangement of leaves on a stem, the spirals of sunflower seeds, the branching patterns of trees, and the structure of pinecones and pineapples. These patterns arise due to optimization principles in nature, making Fibonacci numbers a fundamental aspect of natural growth and organization.

Q. How can Python's built-in libraries, such as NumPy, be utilized to optimize Fibonacci sequence computations?

Python's built-in libraries, such as NumPy, can be leveraged to optimize Fibonacci sequence computations through matrix exponentiation. This basic idea exploits the fact that the nth Fibonacci number can be obtained by raising a specific Fibonacci matrix to the power of n and extracting the appropriate element from the resulting matrix.

Q. What are the applications of the Fibonacci series?

The Fibonacci series finds applications in various fields, including mathematics, computer science, finance, natural sciences, and cryptography. It is used in algorithms, technical analysis in stock trading, modelling natural phenomena, and generating pseudorandom numbers, among other applications.

Q. What are the advantages and disadvantages of using recursion to compute the Fibonacci series in Python?

Using recursion to compute the Fibonacci series in Python offers both advantages and disadvantages.

• The primary advantage is its simplicity and clarity; the recursive Fibonacci function directly mirrors the mathematical definition of the Fibonacci sequence, making the code easy to understand and write.
• This approach is particularly useful for educational purposes, as it clearly illustrates the concept of recursion and the base cases.
• However, the disadvantages are significant, especially for larger integer input values.
• Recursive computation of the Fibonacci series can be highly inefficient due to its exponential time complexity, O(2^n), resulting from redundant calculations of the same subproblems.
• Each Fibonacci number is recalculated multiple times, leading to excessive function calls and increased execution time.
• Additionally, deep recursion can cause a stack overflow, as Python has a limited recursion depth.
• These inefficiencies make the naive recursive solution impractical for large n.
• In such cases, iterative or dynamic programming methods are preferred for generating the Fibonacci series in Python.

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