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Algebra

Sum of Integers 1 to N

n(n+1)/2. Free online Sum of Integers 1 to N. Calculate sum of integers 1 to n online — fast, accurate, mobile-friendly, no signup needed.

Sum
5,050

Derivation

  1. ├── 01Givenn = 100
  2. ├── 02Formulat × (t+1) / 2
  3. └── 03Compute Sum5,050
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§01What is

Understanding the Sum of Integers 1 to N

The Sum of Integers 1 to N computes Sum from 1 input: n. n(n+1)/2.

Algebra is the art of solving for the unknown. Rearranging a formula to isolate the variable you actually need is the single most common real-world math skill — and doing it with real numbers under time pressure is where errors happen. The Sum of Integers 1 to N sits in that toolkit — it n(n+1)/2. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

t × (t+1) / 2

Where

n
N
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: N = 100.

  1. 01Start by noting the input — N: 100.
  2. 02Substitute these values into the formula: t × (t+1) / 2
  3. 03Compute Sum: the calculator returns 5050.
  4. 04Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Sum of Integers 1 to N Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

N halved

n = 50 (from 100)

Keep every other input at its default and halve the n. See how sum responds.

  1. 01New N: 50
  2. 02Baseline Sum: 5050
  3. 03New Sum: 1275
  4. 04Sum decreases by 74.8% → use this sensitivity to plan for real-world variation.
02 · PATTERN

N doubled

n = 200 (from 100)

Keep every other input at its default and double the n. See how sum responds.

  1. 01New N: 200
  2. 02Baseline Sum: 5050
  3. 03New Sum: 20100
  4. 04Sum increases by 298% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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