Sum of Series Calculator — Arithmetic & Geometric
Calculate the partial sum of an arithmetic or geometric series with step-by-step working. For geometric series with |r| < 1 the infinite sum limit is shown automatically.
Series type
Sₙ = n/2 × (a₁ + aₙ)
Formula
Last term aₙ = a₁ + (n−1)d
Sum S8 = 8/2 × (1 + 22)
- 1
Last term aₙ = a₁ + (n − 1) × d
1 + (8 − 1) × 3 = 22 - 2
First plus last term
1 + 22 = 23 - 3
Sum S8 = n ÷ 2 × (a₁ + aₙ)
8 ÷ 2 × 23 = 92
How does this calculator work?
Arithmetic series: Sₙ = n/2 × (a₁ + aₙ). Geometric series: Sₙ = a(1−rⁿ)/(1−r). Infinite geometric (|r|<1): S = a/(1−r). Select series type, enter first term, difference or ratio and term count. Step-by-step working is shown for each calculation.
Formula
How this is calculated
An arithmetic series sums terms that increase by a fixed amount d each step. The partial sum Sₙ = n/2 × (a₁ + aₙ) exploits the symmetry of pairing first and last terms — each pair adds to the same constant — giving an O(1) formula regardless of how large n is.
A geometric series sums terms where each is multiplied by a constant ratio r. The partial sum formula Sₙ = a(1 − rⁿ)/(1 − r) (for r ≠ 1) comes from telescoping: multiplying the sum by r and subtracting cancels all interior terms. When r = 1 every term is identical and Sₙ = n × a. For |r| < 1 the terms shrink toward zero and the infinite series converges to S = a/(1 − r); when |r| ≥ 1 the series diverges and only the finite partial sum is meaningful.
For very large n with |r| > 1 floating-point overflow produces very large numbers or Infinity. The step-by-step panel shows each substitution so you can verify the arithmetic by hand.
Frequently asked questions
When the absolute value of the common ratio is strictly less than 1 (|r| < 1). Each successive term is a fraction of the previous one, so the running total approaches a/(1 − r). If |r| ≥ 1 the series diverges.
A sequence is the ordered list of terms (a₁, a₂, …, aₙ). A series is the running sum of those terms (a₁ + a₂ + … + aₙ). This calculator computes the series sum, not the list.
Yes. Set type = Arithmetic, first term = 1, common difference = 1, number of terms = 100. The result is 5050 — the sum Gauss reportedly computed instantly as a schoolchild.
Also known as
TG we-Calculate Editorial Team. (2026). Sum of Series Calculator — Arithmetic & Geometric [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/sum-of-series-calculator
TG we-Calculate Editorial Team. "Sum of Series Calculator — Arithmetic & Geometric." TG we-Calculate. 2026. https://we-calculate.com/calculator/sum-of-series-calculator.
TG we-Calculate Editorial Team, "Sum of Series Calculator — Arithmetic & Geometric," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/sum-of-series-calculator
@misc{wecalculate_sum_of_series_calculator, title = {Sum of Series Calculator — Arithmetic & Geometric}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/sum-of-series-calculator}}, year = {2026}, note = {TG we-Calculate} }
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