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Calculator Answers

Summation Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A summation calculator computes the total sum of a sequence of terms defined by arithmetic or geometric progressions. In arithmetic sequences, terms increase by a fixed difference, while geometric sequences scale by a common ratio. The calculator applies closed-form algebraic formulas to compute the total sum of the first 'n' terms without requiring manual iteration.

Calculate the sum of arithmetic or geometric sequences. Choose your sequence type, input the starting values, and get the nth term and summation total.

Quick Answer

Find the sum of arithmetic or geometric sequences. Enter the starting terms, sequence difference or ratio, and number of iterations.

Sequence type

What to compute

e.g. 2

e.g. 3

A positive whole number. · e.g. 10

Result

Arithmetic a_n

29

Term formulaa_n = a₁ + (n − 1) × d = 2 + 9 × 3 = 29
Sum formulaS_n = n × (a₁ + a_n) / 2 = 10 × (2 + 29) / 2 = 155
Preview2, 5, 8, 11, 14, 17, 20, 23, …

Arithmetic sequences add the same number each step; geometric sequences multiply by the same ratio each step. Both have closed formulas for the nth term and the sum of the first n terms.

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Examples

Arithmetic sum: a₁ = 1, d = 1, n = 100

Last term a_100 = 100, Sum S_100 = 5,050

Geometric sum: a₁ = 3, r = 2, n = 5

Last term a_5 = 48, Sum S_5 = 93

How it works

Closed-Form Summation Math

This calculator evaluates arithmetic and geometric summations using direct closed-form algebraic formulas:

Arithmetic Sum · S_n = n × (a₁ + a_n) / 2

Geometric Sum · S_n = a₁ × (1 − rⁿ) / (1 − r)

To analyze sequence structures or find individual terms, visit our general series calculator or use our specialized series test tools like the infinite series calculator.

Learn the concept

These guides explain the notation this tool evaluates, with worked examples.

What Is a Sum in Math? Sigma notation, the 1-to-n shortcut, and sum vs product.What Is a Product in Math? The operation a sum is most often confused with.

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Arithmetic Sequence CalculatorFind any specific term (a_n) or the cumulative sum (S_n) of an arithmetic sequence. Features detailed formula substitution steps and an interactive term sequence progression table.

Frequently asked questions

Summation is the addition of a sequence of numbers, resulting in their sum or total. It is represented mathematically using the capital Greek letter sigma (Σ).

Sigma notation is a convenient shorthand representation for writing long sums. It specifies a starting index (below Σ), an ending index (above Σ), and a formula for the terms being added (to the right of Σ).

For an arithmetic series where each term increases by a common difference 'd', the sum is calculated as: S_n = n × (a₁ + a_n) / 2, where 'n' is the number of terms, 'a₁' is the first term, and 'a_n' is the last term.

For a geometric series where each term is multiplied by a common ratio 'r', the sum of the first 'n' terms is calculated as: S_n = a₁ × (1 − rⁿ) / (1 − r), provided r ≠ 1. If r = 1, the sum is simply n × a₁.