Problem
"Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first ten terms will be:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ,...
By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms."
Solution
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. The sequence typically starts with 0 and 1, resulting in the following sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34 , ...
To generate the next number in the sequence, you simply add the two previous numbers. In mathematical terms:
F (n) = F (n-1) + F (n-2)
Where F (n) represents the nth Fibonacci number, F (n-1) is the (n-1)th Fibonacci number, and F(n-2) is the (n-2)th Fibonacci number.
To solve this problem, we will write a Python program that generates Fibonacci numbers and calculates the sum of even numbers up to a given limit.
Here is a python implementation of this:
1. We
define a function Fibonacci_of that
takes a single argument, n,
which represents the maximum value for Fibonacci numbers.
2. We
initialize a and b
with the first two Fibonacci numbers, 1 and 2.
3. The
Sum
variable is initialized to zero, which will store the sum of even Fibonacci
numbers.
4. The
while
loop continues until the current
Fibonacci number exceeds the specified limit.
5. Inside
the loop, we check if the current
Fibonacci number is even (i.e., divisible by 2) using the modulo operator. If
it is, we add it to Sum.
6. We
then update a and b to generate the next
Fibonacci number by shifting them one position forward in the sequence.
7. The
loop continues until b exceeds the limit, and
after the loop, we return the value of Sum,
which contains the sum of even Fibonacci numbers below the limit.
Solving Project Euler Problems provides a great opportunity to practice coding and algorithmic thinking.
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