Discover the Fibonacci series and its significance in mathematics and programming. Learn how to generate the Fibonacci sequence using Python code, understand the mathematical principles behind it.
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Table of Contents
Fibonacci Series
Fibonacci Series is a series in which a number of series is generated by adding two previous numbers. The series starts by default with 0 and 1.
Mathematically,
Fn = Fn-1 + Fn-2
Where Fn denotes the nth term of the Fibonacci series.
The first two terms of this series are:
F0 = 0 (0th term of Fibonacci sequence)
F1 = 1 (1st term of Fibonacci sequence)
Now, by using the above two values we can easily calculate the Fibonacci series:
F2 = F1 + F0 = 0 + 1 = 1
F3 = F2 + F1 = 1 + 1 = 2
F4 = F3 + F2 = 2 + 1 = 3
F5 = F4 + F3 = 3 + 2 = 5
F6 = F5 + F4 = 5 + 3 = 8
F7 = F6 + F5 = 8+ 5 = 13
F8 = F7 + F6 = 13+ 8 = 21
# 0+1=1, 1+1=2, 2+1=3, 3+2=5, 5+3=8, 8+5=13, 13+8=21
E.g., 0, 1, 1, 2, 3, 5, 8,13, 21
Example: Find the Fibonacci series.
List = [0,1]
num1 = 0
num2 = 1
result = num1+num2 #0+1=1
List.append(result) #[0,1,1]
print(List) #[0,1,1]
result = List[2]+List[1] #1+1=2
List.append(result) #[0,1,1,2]
print(List)
result = List[3]+List[2] #1+2=3
List.append(result) #[0,1,1,2,3]
print(List)
Explanation:
The Fibonacci series starts by default with List [0,1] therefore default numbers 0 and 1 defined as num1 and num2 respectively to find the next number.
The variable result is defined as the sum of num1+num2.
The value of the result (num1+num2) is added to the variable List using append() function and displayed using print() function.
Now, List = [0, 1, 1]
To get the next number in the list –
Add first-two previous numbers (from right side) by accessing the elements at the index number.
Here, we calculate the sum of the second element (List[1], which is 1) and the third element (List[2], which is also 1). The result will be 2.
Using while loop
Example: Find the Fibonacci series using a while loop.
n = 8
first_number = 0
second_number = 1
sum = 0
count = 0
while count <= n:
print(first_number)
sum = first_number+second_number
first_number = second_number
second_number = sum
count+=1
#output
[0,1,1,2,3,5,8,13,21]
Explanation:
The variable n represents the number of times to generate the Fibonacci series.
The variable first_number and second_number represent 0 and 1 respectively as the default starting numbers of Fibonacci series.
The code begins with while loop condition count <= n i.e., until the count > n while loop runs.
Inside the loop, the current value of first_number is displayed.
To calculate the next number in the Fibonacci series, the addition of first_number and second_number is stored in the variable sum.
Now, the value of first_number and second_number is swapped. The first_number stores the value of previous second_number and the second_number stores the value of sum.
In each loop, the count value is incremented by 1 to keep track of the iterations.
The loop follows these steps until it reaches the count > n and the Fibonacci series is printed.
Note: if n is 8, the loop will iterate 9 times (count goes from 0 to 8), and the first 9 terms of the Fibonacci series will be printed.
Using def function
Example: Fibonacci series using def function.
Fibo_series = [0,1]
def fibo_sequence(n):
for x in range(2, n):
Next_num = Fibo_series[x-1]+Fibo_series[x-2]
Fibo_series.append(Next_num)
return Fibo_series
print(fibo_sequence(8))
#output:
[0, 1, 1, 2, 3, 5, 8, 13]
#rough explanation of adding two previous numbers and generating the next number in series.
#[2-1] + [2-2] = [1] + [0] = 1+0=1 i.e., [0,1,1]
#[3-1] + [3-2] = [2] + [1] = 1+1=2 i.e., [0,1,1,2]
#....................... i.e., [0,1,1,2,3,5,8]
#[7-1] + [7-2] = [6] + [5] = 8+5=13 i.e., [0,1,1,2,3,5,8,13]
Explanation:
The code begins with the default list of Fibonacci series as Fibo_series[0, 1].
Created a function fibo_sequence that takes a parameter n where the variable n represents the number of times to generate the Fibonacci series.
Inside the function, for loop along with the range() function is used to iterate over the number from 2 to n-1. We started from 2 because we have already initialized the first two numbers (0 and 1) in the Fibo_series.
Now to calculate the next number in the series, we add two previous numbers from the list.
By decrementing x by 1 in the loop, we access the previous number (Fibo_seq[x-1]) required for the addition. Then, by decrementing i by 2 (i-2), we access the number two steps back (Fibo_seq[x-2]) required for the addition.
Access the elements from the Fibo_series through the index number obtained from Fibo_series[x-1]+Fibo_series[x-2] and store the value in the variable Next_num.
Now, Next_num (that is the next number) is added to the Fibo_series using the append() method.
for loop iterate through the numbers until it reaches to the n=8 and repeats all steps. Once the loop completes iterating all numbers, the command reaches to return Fibo_series that stores the all next numbers obtained from Next_num.
Finally, call the function fibo_sequence and declare the value of n to it.
Conclusion
The article provides a comprehensive overview of the Fibonacci series and demonstrates how to generate it using Python programming. It explains the concept of the Fibonacci sequence and its mathematical logic. The article presents a clear and concise Python code example that utilizes a while loop and def function to generate the series.
