The Fibonacci Series Is One Of The Most Important Numerical Series In Mathematics, Which Is Defined Using Its Previous Members. 

Different methods can be used to calculate this series. In this article, we will get to know some of these methods.

Definition of the Fibonacci series

As mentioned, the Fibonacci series is one of mathematics’s most important numerical series, defined using its previous members. To define the Fibonacci series, we first set its first two members equal to 1. Then, each next member of the series equals the sum of the two previous members. In other words:

F0 = 1
F1 = 1
Fn = Fn-1 + Fn-2 (for n ≥ 2)

For example, the elementary members of the Fibonacci series are as follows:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

The Fibonacci series is used in many fields of mathematics, computer science, economics, and physics and has many properties, including being natural, having interesting patterns, and being related to the golden ratio.

What are the uses of the Fibonacci series?

The Fibonacci series is used in many mathematics, computer science, economics, and physics fields. Below I mention some of the applications of this series in different areas:

1.  Population growth: The Fibonacci series simulates population growth well and is used in modeling phenomena such as population growth.
2.  Technical analysis: The Fibonacci series is also used to predict the price trend of financial markets.
3.  Search algorithms: Fibonacci series is used in search algorithms such as the Fibonacci algorithm, reverse Fibonacci algorithm, and Grande Fibonacci algorithm.
4.  Geometric design: In geometric design, the Fibonacci series designs geometric patterns such as stars, linear curves, and aesthetics in artistic designs.
5.  Neural networks: In neural networks, the Fibonacci series is used to design the architecture of neural networks with an optimal structure and high efficiency.
6.  Cryptography: In cryptography, the Fibonacci series generates random numbers by permuting the elements in the series.
7.  Sound and image: The Fibonacci series compresses pictures and sounds and extracts their features in signal processing.

The Fibonacci series is used in many fields as one of the most important numerical series in mathematics and applied sciences.

Calculate and print the Fibonacci series up to the nth term with the repetition loop method

To calculate and print the Fibonacci series up to the nth term using the repetition loop method, you can use the following code in Python:

n = int(input(“Enter the number of terms: “)) # Input the number of members of the series

# The first and second values ​​of the Fibonacci series

first_term = 1

second_term = 1

# Print the first two members of the series

print(first_term)

print(second_term)

# loop to calculate and print the remaining members of the series

for i in range(2, n):

next_term = first_term + second_term # Calculate the next member of the series

print(next_term) # Print the next member of the series

first_term = second_term # Swap members

second_term = next_term

In this code, we first receive the number of series members from the user. Then, we set the first and second values of the Fibonacci series equal to 1 and print them. Then, we calculate and print the next series members using the loop. At each step, the next member of the series is equal to the sum of the two previous members. Finally, the Fibonacci series up to the nth term is printed in order.

Calculate and print the Fibonacci series up to the nth term with the return function method

To calculate and print the Fibonacci series up to the nth term using the return function method, you can use the following code in Python:

def fibonacci(n):

if n <= 1:

return n

Otherwise:

return fibonacci(n-1) + fibonacci(n-2)

n = int(input(“Enter the number of terms: “)) # Input the number of members of the series

# Print the members of the series using the return function

for i in range(n):

print(fibonacci(i))

In this code, we first receive the number of series members from the user. Then we define the Fibonacci return function, which uses the values ​​of the n-1 and n-2 members to calculate the nth member of the series. In this function, if n is less than or equal to 1, the nth term is similar to n, and otherwise, the nth time is calculated using n-1 and n-2 terms. Then, using the loop, the series members up to the nth time are printed using the return function.

For example, to calculate and print the members of the Fibonacci series up to the 10th term, the output of the code will be as follows:

0

1

1

2

3

5

8

13

21

34

It should be noted that using the return function method to calculate the Fibonacci series in cases with a large number of members, due to the increase in the number of function calls, causes the program to run longer, and we may encounter memory problems and unstable returns. Therefore, in cases where many members in the Fibonacci series are required, it is better to use the repeating loop method.

Calculate and print the nth term of the Fibonacci series with the return function method

To calculate and print the nth term of the Fibonacci series using the return function method, you can use the following code in Python:

def fibonacci(n):

if n <= 1:

return n

Otherwise:

return fibonacci(n-1) + fibonacci(n-2)

n = int(input(“Enter the value of n: “)) # input n

# Calculate and print the nth term of the Fibonacci series using the return function

print(“The”, n, “th Fibonacci number is:”, fibonacci(n-1))

In this code, we first get the value of n from the user. Then we define the Fibonacci return function, which uses the importance ​​of the n-1 and n-2 members to calculate the nth member of the series. In this function, if n is less than or equal to 1, the nth term is similar to n, and otherwise, the nth time is calculated using n-1 and n-2 terms. Then, using the return function, the nth time of the Fibonacci series is calculated and printed.

For example, to calculate and print the 10th term of the Fibonacci series, the output of the code will be as follows:

The 10th Fibonacci number is: 55

It should be noted that using the return function method to calculate the Fibonacci series in cases with a large number of members, due to the increase in the number of function calls, causes the program to run longer, and we may encounter memory problems and unstable returns. Therefore, in cases where many members in the Fibonacci series are required, it is better to use the repeating loop method.

Is there a repeating loop method to calculate the Fibonacci series in Python?

Yes, you can also use the iteration loop method in Python to calculate the Fibonacci series. To calculate the Fibonacci series using an iteration loop in Python, you can use the following code:

n = int(input(“Enter the number of terms: “)) # Input the number of members of the series

# The first and second values ​​of the Fibonacci series

first_term = 0

second_term = 1

# Print series members

for i in range(n):

print(first_term)

temp = first_term # store the value of the previous member

first_term = second_term # Swap members

second_term = temp + second_term

In this code, we first receive the number of series members from the user. Then, we set the first and second values of the Fibonacci series equal to 0 and 1 and print the first member of the series. Then, we calculate and print the next series members using the loop. At each step, the next member of the series is equal to the sum of the two previous members. Also, at each step, we store the value of the last member in the temp variable and then swap the members. Using this method, the Fibonacci series up to the nth term is printed in order.

For example, to calculate and print the members of the Fibonacci series up to the 10th term using an iteration loop, the output of the code will be as follows:

0

1

1

2

3

5

8

13

21

34

It should be noted that the repetition loop method for calculating the Fibonacci series in Python is considered the best method for calculating this series due to its higher efficiency.

Another way to calculate the Fibonacci series in Python

Yes, one of the other ways to calculate the Fibonacci series in Python is to use a closed formula. This formula is based on a closed formula for calculating the nth member of the Fibonacci series. To use this formula in Python, you can use the following code:

import math

n = int(input(“Enter the value of n: “)) # input n

# Calculation of the nth term of the Fibonacci series using the closed formula

fibonacci_n = (1 / math.sqrt(5)) * (((1 + math.sqrt(5)) / 2) ** n – ((1 – math.sqrt(5)) / 2) ** n)

# Print the nth term of the Fibonacci series

print(“The”, n, “th Fibonacci number is:”, round(fibonacci_n))

In this code, we first get the value of n from the user. Then, using the closed formula, we calculate the nth term of the Fibonacci series and store it in the Fibonacci_n variable. Finally, using the round command, we convert the result to the nearest integer and print it.

For example, to calculate and print the 10th term of the Fibonacci series using a closed formula, the output of the code will be as follows:

The 10th Fibonacci number is: 55

It should be noted that using a closed formula to calculate the Fibonacci series shortens the execution time of the program. In cases where a large number of members are needed in the Fibonacci series, it is a suitable method to calculate it.

Different methods of calculating the Fibonacci series

In general, the important techniques for estimating the Fibonacci series are as follows:

Analytical method

This method uses a closed formula to calculate the Fibonacci series, and as a result, it is calculated accurately and without errors. The fast formula for calculating the nth member of the Fibonacci series is as follows:

Fn = (1/√5) * [((1+√5)/2)^n – ((1-√5)/2)^n]

Recursive method

In this method, the two previous members are used to calculate the nth member of the Fibonacci series. In other words, the nth member of the sequence equals the sum of its two previous members. This method is suitable for calculating the Fibonacci series, but for large numbers, it may take a lot of time through the processor due to excessive computational repetition.

Circular method

In this method, all the members of the Fibonacci series are stored in an array, and by using the loop, the next members of the series are calculated. Due to the use of more memory, this method is suitable for calculating the Fibonacci series with a large volume.

Matrix method

In this method, the Fibonacci series is defined as a matrix, and the nth member of the series is calculated using the power of the matrix. This method is suitable for calculating the Fibonacci series with large numbers, but due to the computational complexity, it may take a lot of time through the processor.

In this regard, the matrix on the left equals a two-in-two matrix, and the matrix on the right is similar to a binary vector that includes the last two members of the Fibonacci series. To calculate the Fibonacci series using the matrix method in Python, you can use the following code:

Import numpy as np

def fibonacci_matrix(n):

# Define the initial matrix

F = np.array([[1, 1], [1, 0]])

# We calculate the n-1 power matrix of F

Fn = np. Lining.matrix_power(F, n – 1)

# Calculate the nth member of the Fibonacci series

return Fn[0][0]

n = int(input(“Enter the value of n: “)) # input n

# Print the nth term of the Fibonacci series using the matrix method

print(“The”, n, “th Fibonacci number is:”, fibonacci_matrix(n))

In this code, the fibonacci_matrix function is defined to calculate the nth member of the Fibonacci series using the matrix method. First, we describe the initial matrix F using the numpy library. Then, using the np.linalg.matrix_power function, we calculate the n-1 power matrix of the F matrix. Finally, by returning the 0,0 member of the resulting matrix, we produce and print the value of the nth member of the Fibonacci series.

For example, to calculate and print the 10th term of the Fibonacci series using the matrix method, the output of the code will be as follows:

The 10th Fibonacci number is: 55

It should be noted that the matrix method for calculating the Fibonacci series is speedy and efficient due to the computational nature of the matrix. In cases where a large number of members in the series are required, it is a suitable method for calculating it.