Summation Calculator Formula
Understand the math behind the summation calculator. Each variable explained with a worked example.
Formulas Used
Sum Integers
sum_integers = n * (n + 1) / 2Sum Squares
sum_squares = n * (n + 1) * (2 * n + 1) / 6Sum Cubes
sum_cubes = pow(n * (n + 1) / 2, 2)Requested
requested = k == 0 ? n : (k == 1 ? n * (n + 1) / 2 : (k == 2 ? n * (n + 1) * (2 * n + 1) / 6 : pow(n * (n + 1) / 2, 2)))Variables
| Variable | Description | Default |
|---|---|---|
n | Upper Limit (n) | 10 |
k | Power (k) | 2 |
How It Works
Summation Formulas
Sum of Integers
sum(i, 1, n) = n(n+1)/2
Sum of Squares
sum(i², 1, n) = n(n+1)(2n+1)/6
Sum of Cubes
sum(i³, 1, n) = [n(n+1)/2]²
Notice that the sum of cubes equals the square of the sum of integers.
Worked Example
Sum of i² from 1 to 10.
- 01Sum of integers: 10×11/2 = 55
- 02Sum of squares: 10×11×21/6 = 2310/6 = 385
- 03Sum of cubes: (55)² = 3025
Frequently Asked Questions
What is a summation?
A summation (denoted by the sigma symbol) adds up a sequence of terms following a pattern. For example, sum from i=1 to 5 of i² = 1+4+9+16+25 = 55.
Why does the sum of cubes equal the square of the sum?
This is a beautiful identity: 1³+2³+...+n³ = (1+2+...+n)². It can be proven algebraically by induction or geometrically by arranging squares.
Are there formulas for higher powers?
Yes, Faulhaber's formulas give closed-form expressions for the sum of any fixed power k. However, they become increasingly complex for larger k.
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