Summation Formulas
Calculate the sum of arithmetic and geometric series quickly.
Essential shortcuts for adding sequences of numbers.
The Formula
Sum of first n integers: Σk = n(n+1)/2
Sum of first n squares: Σk² = n(n+1)(2n+1)/6
Sum of first n cubes: Σk³ = [n(n+1)/2]²
Sum of first n squares: Σk² = n(n+1)(2n+1)/6
Sum of first n cubes: Σk³ = [n(n+1)/2]²
These closed-form summation formulas let you calculate totals instantly without adding each term one by one. They are building blocks for statistics, calculus, and computer science.
Variables
| Symbol | Meaning |
|---|---|
| Σ | Sigma — summation symbol (add all terms) |
| k | Index variable (runs from 1 to n) |
| n | Number of terms to sum |
Example 1
Find the sum of all integers from 1 to 100
Σk = n(n+1)/2 = 100 × 101 / 2
Sum = 5,050
Example 2
Find the sum of squares from 1² to 10²
Σk² = n(n+1)(2n+1)/6 = 10 × 11 × 21 / 6
= 2,310 / 6
Sum = 385
When to Use It
Use summation formulas when:
- Adding large sequences of numbers efficiently
- Deriving formulas in statistics (variance, standard deviation)
- Analyzing algorithm complexity in computer science
- Solving series and sequences problems in calculus