# sequence

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## sequence

**sequence,**in mathematics, ordered set of mathematical quantities called terms. A sequence is said to be known if a formula can be given for any particular term using the preceding terms or using its position in the sequence. For example, the sequence 1, 1, 2, 3, 5, 8, 13, … (the Fibonacci sequence) is formed by adding any two consecutive terms to obtain the next term. The sequence −1-2, 1, 7-2, 7, 23-2, 17, … is formed according to the formula (

*n*

^{2}− 2)/2 for the

*n*th, or general, term. A sequence may be either finite, e.g., 1, 2, 3, … 50, a sequence of 50 terms, or infinite, e.g., 1, 2, 3, … , which has no final term and thus continues indefinitely. Special types of sequences are commonly called progressions. The terms of a sequence, when written as an indicated sum, form a series; e.g., the sum of the sequence 1, 2, 3, … 50 is the series 1 + 2 + 3 + … + 50.

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Sequence

a fundamental concept of mathematics. A sequence is a set of elements of any nature that are ordered as are the natural numbers 1,2,…, n…. It can be written in the form *x*_{1}, *x*_{2}, …, *x*_{n}, … or simply {*x _{n}*}. The elements of which it is composed are called its terms. Different terms of a sequence may be identical.

A sequence may be regarded as a function whose argument can take on only positive integral values—that is, a function defined on the set of natural numbers. To define a sequence, we can either specify its *n*th term or make use of a recurrence formula, by which each term is defined as a function of preceding terms. Fibonacci numbers, for example, are defined through a recurrence formula. The sequences most often encountered are those of numbers or functions. For example,

(1) 1, 2, …, *n*, …

that is, *x _{n}* =

*n*

If the terms of a sequence of numbers differ by an arbitrarily small amount from the number *a* for sufficiently large *n*, the sequence is said to be convergent, and *a* is called its limit. The limit of a sequence of functions is defined in a similar manner. For example, sequences (2) and (4) are convergent, and their limits are 0 and the function 1/(1 + *x*^{2}), respectively. Sequences that are not convergent are said to be divergent. Sequences (1) and (3) are examples of divergent sequences.

## sequence

[′sē·kwəns]*x*

_{1},

*x*

_{2}… which is indexed by the positive integers; more precisely, a function whose domain is an infinite subset of the positive integers. Also known as infinite sequence.

## sequence

**1.**

**a.**

*Cards*a set of three or more consecutive cards, usually of the same suit

**b.**

*Bridge*a set of two or more consecutive cards

**2.**

*Music*an arrangement of notes or chords repeated several times at different pitches

**3.**

*Maths*

**a.**an ordered set of numbers or other mathematical entities in one-to-one correspondence with the integers 1 to

*n*

**b.**an ordered infinite set of mathematical entities in one-to-one correspondence with the natural numbers

**4.**a section of a film constituting a single continuous uninterrupted episode

**5.**

*Biochem*the unique order of amino acids in the polypeptide chain of a protein or of nucleotides in the polynucleotide chain of DNA or RNA