Education
Summation Calculator
Last updated: June 19, 2026
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
Arithmetic a_n
29
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.
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.
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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.
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