Summation Formulas
Reference for arithmetic series Sn = n(a₁+aₙ)/2 and geometric Sn = a₁(1-rⁿ)/(1-r).
Covers sum of integers n(n+1)/2, squares, cubes, and infinite series.
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
Key Notes
- Gauss derived Σk = n(n+1)/2 at age 9 by pairing the first and last terms: 1+100=101, 2+99=101, ... giving 50 pairs of 101 = 5,050
- These formulas assume the index runs from k=1 to n; for a sum starting at k=m, compute the sum from 1 to n and subtract the sum from 1 to m−1
- The sum of the first n cubes equals the square of the sum of the first n integers: Σk³ = (Σk)² — a remarkable identity
Key Notes
- Sigma notation: Σᵢ₌ₘⁿ f(i): Means "evaluate f(i) at each integer from m to n and add the results." The index i is a dummy variable — the sum's value does not depend on what letter is used. The bounds and function f(i) fully define the sum.
- Key closed-form results: Σᵢ₌₁ⁿ i = n(n+1)/2; Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6; Σᵢ₌₁ⁿ i³ = [n(n+1)/2]². These allow computing the sum of the first n integers (or squares, or cubes) without adding term by term.
- Geometric series: Σᵢ₌₀ⁿ rⁱ = (rⁿ⁺¹−1)/(r−1): For |r| < 1, the infinite series converges: Σᵢ₌₀^∞ rⁱ = 1/(1−r). This underpins present-value annuity formulas, probability generating functions, and digital filter design.
- Linearity: Σ(af(i) + bg(i)) = aΣf(i) + bΣg(i): Constants factor out and sums of sums split. This makes simplification of complex summations much easier and mirrors the linearity of integrals.
- Connection to integrals: A Riemann sum Σf(xᵢ)Δx approximates a definite integral. As n → ∞ and Δx → 0, the sum becomes the integral exactly. This is the fundamental link between discrete summation and continuous integration.