Summation Formula Calculator
Compute closed-form sums: arithmetic series, geometric series, sum of integers, squares, and cubes.
Get the result instantly without adding term by term.
What closed-form summation does
Summation in sigma notation Σ adds a sequence of terms. For many common sequences, you do not have to add the terms one by one. A closed-form formula gives the total directly:
| Sequence | Closed form | Discovered by |
|---|---|---|
| Sum of first n positive integers Σk | n(n+1)/2 | Gauss (anecdotally at age 9, 1786) |
| Sum of first n squares Σk² | n(n+1)(2n+1)/6 | Known to Archimedes, ~250 BCE |
| Sum of first n cubes Σk³ | [n(n+1)/2]² = (Σk)² | Nicomachus, ~100 CE |
| Sum of first n fourth powers Σk⁴ | n(n+1)(2n+1)(3n²+3n−1)/30 | Faulhaber, 1631 |
| Arithmetic series: a₁ + a₂ + … + aₙ | n(a₁ + aₙ)/2 | General formula since antiquity |
| Geometric series: a + ar + ar² + … + arⁿ⁻¹ | a(1 − rⁿ)/(1 − r) for r ≠ 1 | Euclid, ~300 BCE |
| Infinite geometric ( | r | < 1) |
These formulas turn O(n) summations into O(1) constant-time calculations. They are fundamental to combinatorics, calculus (Riemann sums), computer science (loop complexity), and finance (annuities).
The classic Gauss story
The legend: 9-year-old Carl Friedrich Gauss was asked by his teacher to sum the integers from 1 to 100 as busywork. He produced the answer in seconds by pairing 1+100, 2+99, 3+98, and so on, getting 50 pairs of 101 each, for 5050.
Whether the story is exact or embellished, the underlying trick (pairing the first and last terms) is the standard proof of the arithmetic series formula. For 1 to n: pair k with (n+1−k), there are n/2 pairs (or (n−1)/2 plus the middle term for odd n), each summing to n+1. Total = n(n+1)/2.
Arithmetic vs geometric series
An arithmetic series has constant difference between consecutive terms (1, 4, 7, 10, …). The closed form is Sₙ = n(a₁ + aₙ)/2, where aₙ = a₁ + (n−1)d.
A geometric series has constant ratio between consecutive terms (3, 6, 12, 24, …). The closed form is Sₙ = a(1 − rⁿ)/(1 − r). For r = 1 (constant sequence), the sum is just n × a. For |r| < 1, the infinite series converges to a/(1−r), the famous geometric series limit.
The geometric series is everywhere in finance. The present value of a perpetuity, the future value of a series of equal payments, the bond price formula, the dividend discount model: all are closed-form geometric sums or near-relatives.
Worked example: sum of first 50 squares
What is 1² + 2² + 3² + … + 50²?
Use the formula: Σk² = n(n+1)(2n+1)/6 = 50 × 51 × 101 / 6 = 257,550 / 6 = 42,925.
Verify by partial sum check: Σk² for n=10 should be 10·11·21/6 = 385. Hand-add: 1+4+9+16+25+36+49+64+81+100 = 385. Matches.
Worked example: infinite geometric
A bouncing ball loses 30 percent of height on each bounce, starting from 1 meter. Total distance traveled (up + down for each bounce, except the first drop):
Total = 1 (initial drop) + 2 × (0.7 + 0.7² + 0.7³ + …) = 1 + 2 × [0.7 / (1 − 0.7)] = 1 + 2 × 2.333 = 5.667 meters
The infinite series converges to a finite number despite the ball bouncing infinitely many times in principle. This is Zeno’s paradox resolved by calculus.
Why these formulas work
The closed forms are not arbitrary. Each can be proved by induction (assume true for n, prove true for n+1) or by clever algebraic manipulation. The cube-sum identity Σk³ = (Σk)² is especially elegant. The sum of the first n cubes is the square of the sum of the first n integers. There is no obvious geometric reason this should be true; it is a numerical coincidence that has bewitched recreational mathematicians for centuries.
For higher powers (Σk⁴, Σk⁵, etc.), Faulhaber’s formula gives the general pattern. Each Σk^p is a polynomial in n of degree p+1, and the coefficients involve Bernoulli numbers. This connection between summation, Bernoulli numbers, and the Riemann zeta function is one of the threads that ties number theory to analysis.
Telescoping sums
A separate family of closed forms comes from telescoping series. Σ[1/k(k+1)] = Σ[1/k − 1/(k+1)] = 1 − 1/(n+1), because consecutive terms cancel. Telescoping is more of a technique than a formula, but it is in the same spirit: replace an explicit sum with a closed expression.
When closed forms do not exist
Many seemingly simple sums have no elementary closed form. Σ1/k (the harmonic series) grows like ln(n) + γ but has no exact closed form for finite n. Σ1/k² approaches π²/6 (proved by Euler in 1735) but again no simple formula for partial sums. For these, you either use the asymptotic approximation or compute term by term.
This is why “easy” summations are special. The arithmetic, geometric, power-sum, and a handful of telescoping forms cover most cases you meet in practice, but they are exceptional. Most arbitrary sums do not collapse this neatly.
Use cases
In finance: annuity present value uses the geometric series with r = 1/(1+i). Loan payment calculations use the same algebra rearranged.
In computer science: counting basic operations in a nested loop usually produces Σk², Σk, or a geometric sum. Big-O analysis hinges on knowing which closed forms apply.
In physics and engineering: discrete sums approximate integrals. Σk² appears in moments of inertia for chains of particles. Σk³ appears in beam deflection calculations. Geometric sums appear in transmission line impedance, RC filter responses, and digital signal processing.
In statistics: variance and standard-deviation formulas reduce to Σx² and Σx terms. Closed-form summations matter when working with large datasets.