Quick answer
For the first n integers, Σk = n(n + 1) ÷ 2, Σk² = n(n + 1)(2n + 1) ÷ 6 and Σk³ = (n(n + 1) ÷ 2)². For n = 10 the sums are 55, 385 and 3,025.
Enter n. The calculator returns the sum of the integers, their squares and their cubes up to n. Use this sum of series calculator to check your own figures.
Related tools: Slope Calculator: Slope, angle, distance and line equation through two points. LCM Calculator: Least common multiple of two or more numbers. Factor Calculator: List every factor of a positive whole number.
How to use the Sum of Series Calculator
- Enter n (a whole number, between 1 and 1000000).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| n | A whole number | ≥ 1, ≤ 1000000 |
Formula and method
Formula
Σk = n(n+1)/2; Σk² = n(n+1)(2n+1)/6; Σk³ = (n(n+1)/2)²
IEEE-754 double precision; shown to at most 6 decimal places.
Faulhaber closed forms evaluated exactly with BigInt.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: n = 10.
| Result | Value |
|---|---|
| Sum of integers 1…n | 55 |
| Sum of squares | 385 |
| Sum of cubes | 3,025 |
Example 2
Inputs: n = 100.
| Result | Value |
|---|---|
| Sum of integers 1…n | 5,050 |
| Sum of squares | 338,350 |
| Sum of cubes | 25,502,500 |
The first line is the headline result (sum of integers 1…n). Below it the sum of Series Calculator lists sum of squares and sum of cubes, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Gauss's trick
Pair 1 with 10, 2 with 9, and so on: five pairs of 11 give 55. Notice also that the sum of cubes, 3,025, equals the square of the sum, 55².
Range
n can run from 1 to 1,000,000. Sums of squares appear in variance formulas and pyramids of stacked spheres. For n = 100 the sum of integers is 5,050.
How the result changes with n
Holding every other input at the first worked example, this table changes only n. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| n | Sum of integers 1…n | Sum of squares | Sum of cubes |
|---|---|---|---|
| 5 | 15 | 55 | 225 |
| 10 | 55 | 385 | 3,025 |
| 20 | 210 | 2,870 | 44,100 |
| 40 | 820 | 22,140 | 672,400 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- n: entering "0" shows "n must be at least 1." beside that field and no result.
Choosing between the Sum of Series Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
| [Factor Calculator](/math/factor-calculator/) | number | Factors |
| [Quadratic Formula Calculator](/math/quadratic-formula-calculator/) | a, b and c | Roots |
| [Log Calculator](/math/log-calculator/) | number x and base b | log base 2 of 8 |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- n from 1 to 1,000,000
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for n to confirm this.
Frequently asked questions
What is the sum of 1 to 100?
The sum 1 + … + 100 is 100 × 101 ÷ 2 = 5,050.
What is the sum of squares to 5?
1 + 4 + 9 + 16 + 25 = 55, which matches 5 × 6 × 11 ÷ 6.
Who found the formula?
Gauss is credited with the trick as a child.
What does the Sum of Series Calculator calculate?
Sum of the first n integers, squares and cubes. It returns Sum of integers 1…n, Sum of squares and Sum of cubes.
What formula does the Sum of Series Calculator use?
Σk = n(n+1)/2; Σk² = n(n+1)(2n+1)/6; Σk³ = (n(n+1)/2)². Faulhaber closed forms evaluated exactly with BigInt.
What do I need to enter in the sum of Series Calculator?
You enter n.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. n from 1 to 1,000,000.
What is an example of a sum of Series result?
With n 10, the sum of integers 1…n is 55.
Related calculators
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0, including complex roots.
- Log Calculator — Logarithm of a number to any base.
- GCF Calculator — Greatest common factor (greatest common divisor) of two or more numbers.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.