# Summation Calculator

Calculate the sum of arithmetic and geometric series. Enter the first term, common difference or ratio, and number of terms for instant results.

## What this calculates

Calculate the sum of an arithmetic or geometric series in one step. Enter the first term, common difference (or ratio), and the number of terms. For example, the arithmetic series 2 + 5 + 8 + 11 + 14 (first term 2, difference 3, 5 terms) sums to 40.

## Inputs

- **Series Type** — options: Arithmetic Series, Geometric Series — Arithmetic has a common difference; geometric has a common ratio.
- **First Term (a)** — The first term of the series.
- **Common Difference or Ratio** — For arithmetic: the constant difference between terms. For geometric: the constant ratio between terms.
- **Number of Terms (n)** — min 1 — How many terms to sum.

## Outputs

- **Sum (Σ)** — The total sum of the series.
- **Last Term** — The nth (final) term of the series.
- **First Few Terms** — formatted as text — The first several terms of the series.
- **Formula Used** — formatted as text — The summation formula applied.

## Details

**Arithmetic Series**

In an arithmetic series, each term increases by a constant difference d. The sum of n terms is:

**S = n/2 x (first + last)** or equivalently **S = n/2 x (2a + (n-1)d)**

Example: Sum of 1 + 2 + 3 + ... + 100 = 100/2 x (1 + 100) = 5,050. This is the famous result Gauss reportedly discovered as a child.

**Geometric Series**

In a geometric series, each term is multiplied by a constant ratio r. The sum of n terms is:

**S = a(1 - r^n) / (1 - r)** (when r is not equal to 1)

Example: 3 + 6 + 12 + 24 + 48 (first term 3, ratio 2, 5 terms) = 3(1 - 32)/(1 - 2) = 3 x 31 = 93.

**When to Use Each**

- Arithmetic: equally spaced numbers (1, 4, 7, 10, ...) or uniform increments
- Geometric: doubling, halving, exponential growth, compound interest, population models

**Infinite Geometric Series**

If the absolute value of r is less than 1, the infinite geometric series converges: S = a / (1 - r). For example, 1 + 1/2 + 1/4 + 1/8 + ... = 1 / (1 - 0.5) = 2.

## Frequently Asked Questions

**Q: What is the difference between arithmetic and geometric series?**

A: In an arithmetic series, each term differs from the previous by a constant amount (the common difference). In a geometric series, each term is multiplied by a constant factor (the common ratio). For example, 2, 5, 8, 11 is arithmetic (d = 3) while 2, 6, 18, 54 is geometric (r = 3).

**Q: What is the formula for the sum of 1 to n?**

A: The sum 1 + 2 + 3 + ... + n = n(n + 1) / 2. This is an arithmetic series with first term 1, last term n, and n terms. So S = n/2 x (1 + n). For n = 100, the sum is 100 x 101 / 2 = 5,050.

**Q: Can a geometric series sum to infinity?**

A: Yes, but only if the absolute value of the common ratio is less than 1. The infinite sum is S = a / (1 - r). For example, 1 + 0.5 + 0.25 + 0.125 + ... = 1 / (1 - 0.5) = 2. If |r| is 1 or greater, the series diverges and has no finite sum.

**Q: What does sigma notation mean?**

A: The Greek letter sigma (Σ) is shorthand for summation. The expression Σ(i=1 to n) of f(i) means 'add up f(i) for every integer i from 1 to n.' For instance, Σ(i=1 to 5) of i² = 1 + 4 + 9 + 16 + 25 = 55.

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Source: https://vastcalc.com/calculators/math/summation
Category: Math
Last updated: 2026-04-08