What is the formula for the sum of the first n positive integers?

Sum = n(n + 1) / 2. For example, the sum of 1 through 100 is 100 × 101 / 2 = 5,050.

Who discovered this formula?

It's popularly attributed to a young Carl Friedrich Gauss, who reportedly found a shortcut to sum 1 through 100 by pairing numbers from opposite ends of the sequence, though the technique was likely known earlier.

Can I sum a range that doesn't start at 1?

Yes. Enter any starting value and ending value — the calculator subtracts the sum up to (start − 1) from the sum up to end, giving the exact sum of your custom range.

Why does pairing numbers from opposite ends work?

In the sequence 1 to n, pairing the first and last, second and second-to-last, and so on always produces the same sum (n + 1) for each pair, and there are n/2 such pairs, giving n(n + 1)/2 total.

How large a range can this calculator handle?

Because the formula is a direct calculation rather than a loop, it computes sums for ranges up to 1 trillion instantly with no performance penalty.

What is the average of a range of consecutive integers?

For any consecutive integer range, the average always equals the midpoint value: (start + end) / 2. This calculator shows that average alongside the sum.

Does this work for summing squares or cubes instead of plain integers?

No — this calculator sums the integers themselves (1 + 2 + 3 + ...). For sums of squares or cubes, a different closed-form formula is needed.

Where is this formula used in real life?

Beyond being a classic teaching example, this summation appears in combinatorics, in analyzing the runtime of nested loops in computer science, and anywhere a triangular-number pattern shows up.

Sum of Positive Integers Calculator — Add a Range of Whole Numbers Instantly

Our Sum of Positive Integers Calculator adds up every whole number in a range — from a simple '1 to n' sum to any custom range between two positive integers — using the elegant closed-form Gauss formula, so the answer is instant even for enormous ranges. You'll see the total sum, the count of numbers added, the average value, and a step-by-step explanation of the formula applied to your specific range.

Quick Answer

The sum of 1 through n equals n(n + 1) / 2 — the closed-form 'Gauss formula.' This calculator extends it to any starting point, giving you the exact sum of any range of consecutive positive integers instantly, along with the count and average of the range.

Start (m)

End (n)

Enter a start and end value, then click Calculate.

How to Use the Sum of Positive Integers Calculator — Gauss Formula Online

1
Enter the starting integer of your range (use 1 for a classic '1 to n' sum).
2
Enter the ending integer of your range.
3
Click 'Calculate' to instantly get the sum, the count of numbers, and the average.
4
Review the step-by-step formula breakdown showing exactly how the sum was computed.

Why Use Sum of Positive Integers Calculator — Gauss Formula Online?

As the story goes, a young Carl Friedrich Gauss stunned his teacher by instantly summing 1 through 100 — not by adding each number individually, but by pairing the first and last (1 + 100), the second and second-to-last (2 + 99), and so on, noticing every pair adds up to the same total. That insight gives the formula n(n + 1) / 2, which computes any such sum in a single step regardless of how large n is. Our calculator applies that same formula (extended to handle any starting point, not just 1) so you get an exact answer instantly, even for ranges spanning millions or billions of numbers.

Frequently Asked Questions

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