Number Sequence Calculator

What is a Number Sequence?

A number sequence is an ordered list of numbers that follow a particular pattern or rule. Each number in a sequence is called a term, and the position of a term is typically denoted by n. Number sequences are fundamental in mathematics, appearing in everything from simple counting patterns to complex mathematical models used in physics, engineering, and computer science. Understanding sequences helps in recognizing patterns, making predictions, and solving real-world problems. Explore related number concepts with the Multiples Calculator, Multifactorial Calculator, and Number Word to Standard Notation Converter.

Types of Number Sequences

There are many types of number sequences, but three of the most common and important ones are arithmetic sequences, geometric sequences, and Fibonacci sequences. Each type follows a distinct rule for generating its terms:

• Arithmetic Sequence: Each term is obtained by adding a fixed constant (common difference) to the previous term. Example: 1, 3, 5, 7, 9, ... where the common difference is 2.
• Geometric Sequence: Each term is obtained by multiplying the previous term by a fixed constant (common ratio). Example: 2, 4, 8, 16, 32, ... where the common ratio is 2.
• Fibonacci Sequence: Each term is the sum of the two preceding terms, starting with F(0) = 0 and F(1) = 1. Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...

Arithmetic Sequence Formula

An arithmetic sequence follows a linear pattern where each term increases or decreases by the same amount. The nth term of an arithmetic sequence is given by the formula:

aₙ = a₁ + (n - 1) × d

Where a₁ is the first term, d is the common difference, and n is the term number. For example, if a₁ = 3 and d = 4, then the 10th term would be a₁₀ = 3 + (10 - 1) × 4 = 3 + 36 = 39. The sum of the first n terms can be calculated using Sₙ = n/2 × (2a₁ + (n-1)d).

Geometric Sequence Formula

A geometric sequence follows an exponential pattern where each term is multiplied by the same factor. The nth term of a geometric sequence is given by:

aₙ = a × r^(n-1)

Where a is the first term, r is the common ratio, and n is the term number. For example, if a = 2 and r = 3, then the 5th term would be a₅ = 2 × 3⁴ = 2 × 81 = 162. The sum of the first n terms uses Sₙ = a(1-rⁿ)/(1-r) when r ≠ 1.

Fibonacci Sequence

The Fibonacci sequence is one of the most famous sequences in mathematics, named after the Italian mathematician Leonardo Fibonacci. Each term is the sum of the two preceding ones. The sequence often appears in nature, art, and architecture, from the spiral arrangement of sunflower seeds to the proportions of the Parthenon.

F(n) = F(n-1) + F(n-2)

With initial values F(0) = 0 and F(1) = 1. For instance, F(10) = 55, and as the index increases, the ratio between consecutive terms approaches the golden ratio (~1.618).