Definition of Sequence in Mathematics

Introduction of Sequence in Mathematics:

A set of numbers ${s_1},{s_2},{s_3}, \ldots ,{s_n}, \ldots$ in a definite order of occurrence is defined as a sequence in mathematics. It is denoted by $\{ {s_n}\}$ or $\left\langle {{s_n}} \right\rangle$ , where ${s_n}$ is the n^th term of the sequence.

The order of terms is of importance in a sequence. Thus the sequence $\left\{ {1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}, \ldots } \right\}$ is different from the sequence $\left\{ {\frac{1}{2},1,\frac{1}{4},\frac{1}{3},\frac{1}{5}, \ldots } \right\}$ , even though both contain the same terms.

A sequence is actually a function of the natural number n that is written in the natural order, and whose domain is only the set of natural numbers.

Definition of Sequence:

Let S be any non-empty set. A function whose domain is the set N of natural numbers and whose range is a subset of S, is called a sequence in the set S.

The other definition for the sequence in mathematics is; “a sequence in a set S is a rule which assigns to each natural number a unique element of S”. That is, a sequence is a function that maps natural numbers (the sequence’s positions) to the elements at each position.

Sequences of Real Numbers in Mathematics:

A sequence whose range is a subset of R is called a real sequence or a sequence of real numbers. A sequence in R is a function from N into R.

A real sequence is a real-valued function $f:N \to R$ defined on the set N. This number f(n) is the n^th term of this sequence. In mathematics, the n^th term f(n) of a sequence is denoted by $s_n$ or $t_n$ , while the sequence f is denoted by $\{ {s_n}\}$ or $\{ {t_n}\}$ , respectively.

Representation of Sequence:

A sequence in mathematics can be described in several different ways.

(1). Listing in order, the first few elements of a sequence, till the rule for writing down different elements become clear.

For example, $\left\langle {1,8,27,64, \ldots } \right\rangle$ is the sequence whose n^th term is ${n^3}$ .

(2). Defining a sequence by a formula for its n^th term.

For example, the sequence $\left\langle {1,8,27,64, \ldots } \right\rangle$ can also be written as $\left\langle {1,8,27, \ldots ,{n^3}, \ldots } \right\rangle$ or $\left\langle {{n^3}:n \in N} \right\rangle$ or simply $\left\langle {{n^3}} \right\rangle$ .

(3). Defining a sequence by a Recursion formula, that is, by a rule which expresses the n^th term in terms of the ${(n - 1)^{th}}$ term.

For example, let ${a_1} = 1,{a_{n + 1}} = 3{a_n}$ , for all $n \ge 1$ .

These relations define a sequence whose n^th term is ${3^{n - 1}}$ .

Examples of Sequence:

$\left\langle {1,8,27, \ldots ,{n^3}, \ldots } \right\rangle$ or $\left\langle {{n^3}} \right\rangle$ , for all ${n \in N}$ is a sequence.
$\left\{ {\frac{1}{n}} \right\}$ is a sequence $\left\{ {1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}, \ldots } \right\}$ .
$\left\{ {\frac{n}{{n + 1}}} \right\}$ is a sequence $\left\{ {\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}, \ldots } \right\}$ .
$\left\{ {{n^3}} \right\}$ is a sequence $\left\{ {1,8,27, \ldots ,{n^3}, \ldots } \right\}$ .
$\left\{ {\frac{{{{( - 1)}^n}}}{n}} \right\}$ is a sequence $\left\{ { - 1,\frac{1}{2}, - \frac{1}{3},\frac{1}{4}, - \frac{1}{5}, \ldots } \right\}$ .
$\left\{ {{{( - 1)}^n}} \right\}$ is a sequence $\left\{ { - 1,\;1, - 1,\;1, - 1,\;1, \ldots } \right\}$ .

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About Lata Agarwal 270 Articles

M.Phil in Mathematics, skilled in MS Office, MathType, Ti-83, Internet, etc., and Teaching with strong education professional. Passionate teacher and loves math. Worked as a Assistant Professor for BBA, BCA, BSC(CS & IT), BE, etc. Also, experienced SME (Mathematics) with a demonstrated history of working in the internet industry. Provide the well explained detailed solutions in step-by-step format for different branches of US mathematics textbooks.