What is a partial sum?

A partial sum is the sum of the first N terms of a sequence. For an infinite series, the limit of the partial sums (as N → ∞) determines whether the series converges.

What formula is used for an arithmetic series?

S(n) = n/2 × (2a + (n − 1)d), where a is the first term, d is the common difference, and n is the number of terms. Equivalently, S(n) = n × (first + last) / 2.

What formula is used for a geometric series?

S(n) = a × (1 − rⁿ) / (1 − r) when r ≠ 1. If r = 1, S(n) = a × n. Here a is the first term and r is the common ratio.

How many custom values can I enter?

You can paste up to 500 comma-separated numbers. The N field caps at the count of valid numbers in your list.

What if my geometric ratio is greater than 1?

The series diverges — each term grows larger. The calculator still computes the finite partial sum correctly for any ratio you enter.

Partial Sum Calculator

Calculate the sum of the first N terms of an arithmetic sequence, geometric sequence, or any custom list of numbers.

Sequence Settings

Sequence Type

First Term (a₁)

Common Difference (d)

Number of Terms (N)

Term Breakdown

n	Term aₙ	Running Sum S(n)

Set your sequence parameters and click Calculate.

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Summary

Calculate the sum of the first N terms of an arithmetic sequence, geometric sequence, or any custom list of numbers.

How it works

Choose a sequence type: arithmetic, geometric, or custom.
For arithmetic sequences, enter the first term and common difference.
For geometric sequences, enter the first term and common ratio.
For custom sequences, paste a comma-separated list of numbers.
Set how many terms (N) to include in the partial sum.
The calculator instantly shows the sum, each term, and a running total.

Use cases

Verify hand-calculated series sums for homework or exam prep.
Explore how quickly a geometric series converges or diverges.
Sum a custom dataset of values without a spreadsheet.
Check the arithmetic series formula S = n/2 * (2a + (n-1)d) against real numbers.
Find the total distance in motion problems with constant acceleration.
Calculate compound-growth totals over N periods with a geometric series.

Frequently Asked Questions

Last updated: 2026-09-25 · Reviewed by Nham Vu