Fibonacci Sequence

The Fibonacci Sequence is a series of numbers where each number F(n)  is the sum of the two preceding ones, often starting with 0 and 1. 
That is: 

F(n)=F(n−1)+F(n−2)

With initial conditions:

F(0)=0,F(1)=1

Visual Representation

Recursive Approach:

def fibonacci_recursive(n)
  return n if n <= 1
  return fibonacci_recursive(n - 1) + fibonacci_recursive(n - 2)
end

The recursive method is straightforward and mirrors the mathematical definition of the Fibonacci sequence. When n is 0 or 1, the function returns n. Otherwise, it recursively calls itself with n - 1 and n - 2 and adds the results together.

While this approach is elegant and easy to understand, it has a drawback. As n increases, the number of redundant calculations grows exponentially, leading to performance issues for larger values of n.

Iterative Approach:

def fibonacci_iterative(n)
  return n if n <= 1

  prev, curr = 0, 1

  (2..n).each do
    prev, curr = curr, prev + curr
  end

  curr
end

This is an iterative approach to calculating Fibonacci numbers. It uses two variables, prev and curr, to keep track of the last two Fibonacci numbers. It starts from the third Fibonacci number (index 2) and iterates up to n, updating prev and curr in each iteration by shifting them forward in the sequence.
This approach is more efficient than the recursive one, as it avoids redundant calculations. It's a preferred choice for larger values of n due to its linear time complexity.

Performance Considerations:
While the recursive approach is elegant and intuitive, it suffers from performance issues as n increases. Each recursive call results in redundant calculations, slowing down the computation.

On the other hand, the iterative approach is more efficient for larger values of n. It avoids the pitfalls of redundant calculations and achieves linear time complexity, making it the preferred choice when performance matters.

Conclusion:
Both the recursive and iterative approaches provide a way to calculate Fibonacci numbers in Ruby, but they differ significantly in terms of performance. The recursive approach is elegant but inefficient for large inputs, while the iterative approach is more practical and performs well even with substantial values of n.

When working with Fibonacci numbers in real-world scenarios, it's essential to choose the method that aligns with the performance requirements of the application. Understanding the strengths and weaknesses of each approach allows one to make informed decisions based on the specific needs of their projects.