L1 · FoundationsNumbers & loops~3 min · 5 tests

Sum of the First N Natural Numbers

Add 1 + 2 + … + n with a loop in O(n), then with Gauss's formula n(n+1)/2 in O(1). Your first time-complexity lesson, measured live in the browser.

The problem

Return 1 + 2 + … + n for a non-negative integer n. Try to do it without a loop.

Examples

Example 1
Input
```
sum_to_n(10)
```
Expected output
```
55
```
Example 2
Input
```
sum_to_n(1)
```
Expected output
```
1
```

+ 3 hidden tests on Submit.

Edge cases to ask about

n = 0
Large n

Hints

0/3

How an interviewer scores this

0/9

1. Understand the problemRestate it, ask about edge cases and input size.
2. Write a brute-force solutionCorrect first, clever later.
3. Get the correct outputautoAll visible examples pass.
4. Analyse time complexityautoCommit to a Big-O in the Complexity Lab.
5. Analyse space complexityautoExtra memory, not the input.
6. Identify the bottleneckautoMeasure your code — where does it bend?
7. OptimiseUsually: a set or dict instead of a scan.
8. Explain why it worksSay it out loud to Solvi.
9. Test the edge casesautoSubmit passes every hidden test.

Python 3.13 · sum_to_n

def sum_to_n(n):
    """Return 1 + 2 + ... + n."""
    # Write your solution here
    pass

⌘/Ctrl + Enter runs the examples

Your code runs in real CPython inside your browser — nothing is sent anywhere. The first run downloads the interpreter (about 6 MB, once). Your code is saved on this device as you type.

Complexity Lab

What does this cost as n grows?

Interviewers score the analysis as much as the code. Commit to an answer first — then check it, and measure your code against the optimal one at growing input sizes.

Time complexity of the optimal solution

Space complexity (extra memory)

Pick both to reveal the answer.

Measure it

Runs the function on inputs of size 250 up to 16,000 and records the time and peak memory. Slow solutions stop early — a short curve is itself the answer.

From brute force to optimal

The progression an interviewer wants to hear, one step at a time.

Approach	Time	Space	Idea
Loop and add	O(n)	O(1)
bestGauss's formula	O(1)	O(1)	Pair 1 with n, 2 with n−1, …: n/2 pairs each summing to n+1.

Walkthrough of the optimal approach (try it yourself first)

Pair the numbers from both ends: 1 + n, 2 + (n−1), … each pair sums to n + 1 and there are n / 2 pairs, so the total is n(n + 1) / 2.

Run both versions in the Complexity Lab: the loop's time grows in a straight line with n, the formula stays flat. That flat line is what O(1) means.

Complexity: O(1) time, O(1) space. The formula costs the same three operations for any n; the loop grows linearly with n.

Reveal the reference solution

def sum_to_n(n):
    return n * (n + 1) // 2

The brute force, for comparison

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

Follow-ups interviewers ask

Sum of squares 1² + … + n² in O(1).

Frequently asked interview questions

Core interview concepts, complexities, and follow-ups scored by hiring teams.

What is the time complexity of Sum of the First N Natural Numbers in Python?

The optimal solution runs in O(1) time and O(1) auxiliary space. The formula costs the same three operations for any n; the loop grows linearly with n.

What is the brute-force approach, and how do you optimise it?

Loop and add: O(n) time, O(1) space. Gauss's formula: O(1) time, O(1) space. Pair 1 with n, 2 with n−1, …: n/2 pairs each summing to n+1.

What follow-up questions do interviewers ask about Sum of the First N Natural Numbers?

Sum of squares 1² + … + n² in O(1).