Sequence

A sequence is a mathematical term referring to an ordered list of elements, often numbers, but also potentially other mathematical objects like functions or sets. The order of the elements is crucial; changing the order creates a different sequence. Sequences can be finite, meaning they have a definite beginning and end, or infinite, extending without limit.

Sequences are formally defined as functions whose domain is either the set of natural numbers (for infinite sequences) or a finite subset of the natural numbers (for finite sequences). This means each element in the sequence is associated with a specific position or index. The elements of a sequence are often called terms.

Different types of sequences exist, categorized by the relationship between successive terms. Common types include:

• Arithmetic Sequences: Where the difference between consecutive terms is constant (the common difference).
• Geometric Sequences: Where the ratio between consecutive terms is constant (the common ratio).
• Fibonacci Sequence: Where each term is the sum of the two preceding terms.

Sequences are fundamental in various areas of mathematics, including calculus (especially in the study of limits and convergence), analysis, and number theory. They are used to model and analyze patterns, growth, and other phenomena in diverse fields such as physics, computer science, and economics.

In computer science, the term "sequence" is used more broadly to refer to any ordered collection of items, such as a sequence of characters in a string, a sequence of instructions in a program, or a sequence of events in a log file. The core concept of order is still essential, and the elements in the sequence are accessed based on their position.