What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant value is called the 'common difference' (d). Example: 5, 10, 15, 20... (where d=5).

What is a geometric sequence?

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the 'common ratio' (r). Example: 3, 9, 27, 81... (where r=3).

Is the Fibonacci sequence arithmetic or geometric?

Neither. The Fibonacci sequence (0, 1, 1, 2, 3, 5...) is a 'recursive' sequence, where the next term is the sum of the previous two. It does not have a constant difference (1-1=0, 2-1=1) or a constant ratio (2/1=2, 3/2=1.5).

Sequence Calculator

Analyze any number sequence to find the formula, next terms, and sum. Our tool automatically detects **Arithmetic**, **Geometric**, and **Quadratic** sequences.

Enter Your Number Sequence:

Enter at least 3 numbers, separated by commas or spaces.

How to Find the Rule for a Sequence

This calculator is a powerful tool that automatically checks for the most common types of mathematical sequences. Here are the formulas it uses:

1. Arithmetic Sequence

A sequence is **arithmetic** if the difference between terms is constant. This is called the "common difference" (d).

Example: 5, 8, 11, 14... (The common difference `d` is 3).
Explicit Formula (nth term): `aₙ = a₁ + (n-1)d`
Sum Formula (Series): `Sₙ = n/2 * (2a₁ + (n-1)d)`

2. Geometric Sequence

A sequence is **geometric** if the ratio between terms is constant. This is called the "common ratio" (r).

Example: 3, 6, 12, 24... (The common ratio `r` is 2).
Explicit Formula (nth term): `aₙ = a₁ * rⁿ⁻¹`
Sum Formula (Series): `Sₙ = a₁ * (1 - rⁿ) / (1 - r)`

3. Quadratic Sequence

A sequence is **quadratic** if the *second* difference (the difference between the differences) is constant. The formula is in the form of a quadratic equation.

Example: 1, 4, 9, 16... (The sequence of perfect squares).
Explicit Formula (nth term): `aₙ = An² + Bn + C`

Frequently Asked Questions (FAQ)

What is the difference between an arithmetic and geometric sequence?

An **arithmetic** sequence has a *common difference* (you add or subtract the same number each time). A **geometric** sequence has a *common ratio* (you multiply or divide by the same number each time).

Is the Fibonacci sequence (1, 1, 2, 3, 5) arithmetic or geometric?

Neither. The Fibonacci sequence is a **recursive** sequence, where the next term is the sum of the two preceding ones. It does not have a constant difference (1-1=0, 2-1=1) or a constant ratio (2/1=2, 3/2=1.5). Our calculator does not currently detect recursive sequences like Fibonacci.