Mathematics · Algebra

Sum from 1 to N Calculator

Add every whole number from one through n.

Runs locally
Your numbers

Inputs and results stay in this browser. Change one value at a time to explore the relationship.

Your inputCalculatedPassed forward in chains
Sum5,050
Terms100

Calculation steps

  1. 100×(100+12=5050.
  2. Report Sum=5050, Terms=100.

Understand Sum 1 to n

One idea, three depths

Choose how deeply to explain Sum 1 to n

Sum 1 to n: Add every whole number from one through n.

Age 5Explain it to a 5-year-oldStart with a picture

Imagine using Sum 1 to n to answer this question: add every whole number from one through n? Enter Final integer n; the calculator shows Sum. For example: 1+…+100=5050. The answer tells you Sum.

Age 15Explain it to a 15-year-oldConnect it to the formula

Gauss's pairing argument makes the arithmetic sum constant-time. The rule is Σₖ₌₁ⁿk=n(n+1)/2. Its input values are Final integer n, and the main result is Sum. For example: 1+…+100=5050.

CollegeExplain it at college levelState the model precisely

This calculator evaluates the stated sum 1 to n relation over the valid integer domain stated below. The implemented relation is Σₖ₌₁ⁿk=n(n+1)/2, evaluated from Final integer n to produce Sum. Gauss's pairing argument makes the arithmetic sum constant-time. The endpoints 1 and n are both included.

Inputs and valid domain

  • Final integer n must be an integer, at least 1.

Important boundary: The endpoints 1 and n are both included.

The formula

Σₖ₌₁ⁿk=n(n+1)/2

How the calculator works through it

It substitutes Final integer n into the formula and exposes every numerical step above. The main output is Sum, accompanied by Terms.

Read the result correctly

The Sum is the direct answer to “add every whole number from one through n.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.

A worked check

1+…+100=5050.

Where this model stops being reliable

The endpoints 1 and n are both included.

Learn it by changing one value

Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.

Dictionary terms behind this calculator

Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.

These foundations help you understand why Sum 1 to n works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.

Hard requirements

  • Reading formulas and substituting values

    Sum 1 to n uses Σₖ₌₁ⁿk=n(n+1)/2. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.

    Review this foundation about 4 min

Strong support

Optional enrichment

Learn the missing foundationsI already know these — show the code

Mathematics → algorithm → program

Implement this calculation in code

These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.

Algorithm

  1. Read Final integer n.
  2. Evaluate the principal relationship: Σₖ₌₁ⁿk=n(n+1)/2.
  3. Return Sum and check the domain conditions described above.
Python
            from math import *

def sum_one_to_n(n) -> float:
    return ((n * (n + 1.0)) / 2.0)

assert abs(sum_one_to_n(100) - 5050) < 1e-6 * max(1.0, abs(5050))
          
Current calculator valuesUpdates when you change an input above.
              
            
C
            #include <assert.h>
#include <math.h>

double sum_one_to_n(double n) {
    return ((n * (n + 1.0)) / 2.0);
}

int main(void) {
    const double expected = 5050;
    const double actual = sum_one_to_n(100);
    assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
          
Current calculator valuesUpdates when you change an input above.
              
            
C++
            #include <cassert>
#include <cmath>
#include <numbers>

double sum_one_to_n(double n) {
    return ((n * (n + 1.0)) / 2.0);
}

int main() {
    constexpr double expected = 5050;
    const double actual = sum_one_to_n(100);
    assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
          
Current calculator valuesUpdates when you change an input above.
              
            
Linux x86-64 assembly

x86-64 NASM · System V ABI · Linux · SSE2 with libm where required

            ; double sum_one_to_n(double n)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global sum_one_to_n
section .text

sum_one_to_n:
    push rbp
    mov rbp, rsp
    sub rsp, 48
    movsd [rbp-8], xmm0
    mov rax, 0x3ff0000000000000
    movq xmm0, rax
    movsd [rbp-40], xmm0
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-40]
    movsd [rbp-32], xmm0
    movsd xmm0, [rbp-8]
    mulsd xmm0, [rbp-32]
    movsd [rbp-24], xmm0
    mov rax, 0x4000000000000000
    movq xmm0, rax
    movsd [rbp-48], xmm0
    movsd xmm0, [rbp-24]
    divsd xmm0, [rbp-48]
    movsd [rbp-16], xmm0
    movsd xmm0, [rbp-16]
    leave
    ret
          
Current calculator valuesUpdates when you change an input above.
              
            
MATLAB
            function result = sum_one_to_n(n)
    result = ((n * (n + 1.0)) / 2.0);
end
          
Current calculator valuesUpdates when you change an input above.
              
            
Wolfram Language
            ClearAll[mwCalculate];
mwCalculate[n_] := ((n * (n + 1.0)) / 2.0);
          
Current calculator valuesUpdates when you change an input above.
              
            

Continue in mathematical software

The downloaded file includes your current inputs and first calculated result. It is created locally.

Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.

Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations

Standards, reading and academic references

Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.

Algebra and Trigonometry 2e

Read the related free OpenStax mathematics chapters
Cite this book
APA 7
Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
MLA 9
Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
Chicago author-date
Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.

OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.

Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS

These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.

APA 7

MW SysArc. (2026, July 21). Sum from 1 to N Calculator. MW SysArc Tools. https://math.mwsysarc.com/algebra/sum-one-to-n

MLA 9

MW SysArc. “Sum from 1 to N Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/algebra/sum-one-to-n. Accessed 31 Aug. 2026.

Chicago 17

MW SysArc. “Sum from 1 to N Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/algebra/sum-one-to-n.

Harvard

MW SysArc (2026) ‘Sum from 1 to N Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/algebra/sum-one-to-n (Accessed: 31 August 2026).

BibTeX and RIS records

BibTeX

@misc{mwsysarc_sum_one_to_n_2026,
  author = {{MW SysArc}},
  title = {Sum from 1 to N Calculator},
  howpublished = {MW SysArc Tools},
  year = {2026},
  url = {https://math.mwsysarc.com/algebra/sum-one-to-n},
  note = {Published July 21, 2026; accessed August 31, 2026}
}

RIS

TY  - ELEC
AU  - MW SysArc
TI  - Sum from 1 to N Calculator
T2  - MW SysArc Tools
PY  - 2026
DA  - 2026-07-21
Y2  - 2026-08-31
UR  - https://math.mwsysarc.com/algebra/sum-one-to-n
N1  - Published July 21, 2026
ER  -

Clear answers

Frequently asked questions

What does the Sum 1 to n do?

Add every whole number from one through n.

How does the Sum 1 to n work?

The calculator applies Σₖ₌₁ⁿk=n(n+1)/2. Gauss's pairing argument makes the arithmetic sum constant-time.

What can I learn from the Sum 1 to n?

It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.

Does MW SysArc receive or store what I enter?

No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.

How should I use the result?

Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.

Last reviewed . Calculations tested .

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