Summation Calculator: Sigma Notation Sum & Formula

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Formula at a glance

Summation Calculator: Sum of the first n numbers — Σk = n(n + 1)/2. Example: Σ k² from 1 to 10 → Sum: Σ n=1→10 n² = 385 — How the Summation Calculator works: Sum of the first n numbers — Σk = n(n + 1)/2

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📘 About the Summation Calculator

Sigma notation Σ is shorthand for adding a pattern of terms: Σ from k = 1 to 10 of k² means 1² + 2² + … + 10² = 385. Sums appear in sequences and series, statistics (means and variances), finance (repeated deposits) and computer science (counting loop steps).

Type the expression, the index letter and the limits. The calculator adds the terms exactly (fractions stay fractions), lists the first terms, draws the running total and — for polynomial expressions — finds the closed formula for the sum up to n, like n(n + 1)(2n + 1)/6 for Σk².

🧮 Formula

Sum of the first n numbers

Σk = n(n + 1)/2

Sum of squares

Σk² = n(n + 1)(2n + 1)/6

Sum of cubes

Σk³ = [n(n + 1)/2]²

✏️ Worked examples

Σ k² from 1 to 10

You enter
Expression	n^2
Variable	n
From (lower limit)	1
To (upper limit)	10

You get
Sum	Σ n=1→10 n² = 385
Number of terms	10
Expanded	1 + 4 + 9 + 16 + 25 + 36 + …
Formula for the first terms up to n	Σ = n(n + 1)(2n + 1)/6

Σ n=1→10 n² = 385.

Telescoping Σ 1/(n(n + 1))

You enter
Expression	1/(n(n+1))
Variable	n
From (lower limit)	1
To (upper limit)	20

You get
Sum	Σ n=1→20 1/(n(n + 1)) = 20/21 ≈ 0.9523809524
Number of terms	20
Expanded	1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42 + …

Σ n=1→20 1/(n(n + 1)) = 20/21 ≈ 0.9523809524.

🧭 How to use this calculator

What you enter

Expression
VariableOne letter. Other letters are treated as constants.
From (lower limit)
To (upper limit)

What you get

Sum
Number of terms
Expanded
Formula for the first terms up to n

❓ Frequently asked questions

What is the sum of 1 to 100?

5,050 = 100 × 101 ÷ 2 — the trick Gauss is said to have used as a schoolboy.

How do I read sigma notation?

Σ from i = a to b of f(i) means substitute every whole number i from a to b into f and add the results.

What is a telescoping sum?

One where most terms cancel, like Σ 1/(n(n + 1)) = Σ (1/n − 1/(n + 1)) = 1 − 1/(N + 1).

Σ Summation Calculator (Sigma Notation Σ)