Runtime Complexity of Addup Function

Given a function named addup, which adds numbers up to a certain number depending on the argument passed, determine the runtime complexity of the function both for the scenario where it iterates with a for-loop and where it computes using the formula sum = $n \times (n + 1) / 2$.

In this problem, we're tasked with analyzing the runtime complexity of a function 'addup' which involves two different approaches: using a for-loop iteration and applying a mathematical formula. This exercise is a classic example of comparing algorithmic efficiency based on different methods of problem-solving.

The for-loop approach represents an iterative method that computes the sum by accumulating values in a sequential manner. When analyzing this method, it's crucial to assess the number of iterations it requires, which will give you a linear runtime complexity, commonly denoted as O(n). This is because the loop runs n times, where n is the input size, performing a constant amount of work in each iteration.

Conversely, the formula approach exploits a well-known arithmetic series formula to calculate the sum in constant time, regardless of the size of n. This represents an approach that avoids iteration by utilizing a closed-form expression, leading to a constant time complexity of O(1). The formula method is generally more efficient for this specific problem since it reduces the computational overhead associated with loop-based summation.

Overall, this problem highlights the importance of selecting appropriate strategies to optimize performance, a key concept in algorithm design and analysis. Understanding the difference between linear and constant time complexities is fundamental for recognizing how algorithmic choices impact resource usage in terms of time and space.