What is the time complexity of calculating the sum of the first n natural numbers using a loop?Group of answer choicesO(n^2)O(1)O(log n)O(n)

Break Down the Problem

1. We need to analyze the algorithm that calculates the sum of the first $n$ natural numbers using a loop.
2. We will consider how the loop operates and how the running time scales with the input size $n$.

Relevant Concepts

1. The formula to calculate the sum of the first $n$ natural numbers is: $S = 1 + 2 + ... + n = \frac{n(n + 1)}{2}$ However, since we are required to calculate this using a loop, we will loop from 1 to $n$ and add each number to a total.

Analysis and Detail

1. In a loop that runs from 1 to $n$, each iteration represents one addition operation:

   total = 0
   for i in range(1, n+1):
       total += i
   

2. The loop iterates $n$ times, performing a constant amount of work (the addition) in each iteration.

3. Therefore, the total time taken by the loop is proportional to $n$.

Verify and Summarize

1. The time complexity of a single iteration is $O(1)$.

2. Since we perform $n$ iterations, the overall time complexity will be:

   $O(n) \text{ (as the total work done is linear in relation to } n\text{)}$

Final Answer

The time complexity of calculating the sum of the first $n$ natural numbers using a loop is $O(n)$.