Number patterns (sequences)

Number patterns (sequences)

Video: Number Patterns

Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first n natural numbers (for a sequence of finite length n). The position of an element in a sequence is its rank or index; it is the integer from which the element is the image. It depends on the context or of a specific convention, if the first element has index 0 or 1. When a symbol has been chosen for denoting a sequence, the nth element of the sequence is denoted by this symbol with n as subscript; for example, the nth element of the Fibonacci sequence is generally denoted F_n.

Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...). 

Examples and notation

A sequence can be thought of as a list of elements with a particular order. In particular, sequences are the basis for series, which are important in equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences.

Listing is most useful for infinite sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples.

Examples

A tiling with squares whose sides are successive Fibonacci numbers in length.

For a large list of examples of integer sequences, see On-Line Encyclopedia of ......

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