Definition of Sequence in Mathematics
Introduction of Sequence in Mathematics:
A set of numbers in a definite order of occurrence is defined as a sequence in mathematics. It is denoted by or , where is the nth term of the sequence.
The order of terms is of importance in a sequence. Thus the sequence is different from the sequence , 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 defined on the set N. This number f(n) is the nth term of this sequence. In mathematics, the nth term f(n) of a sequence is denoted by or , while the sequence f is denoted by or , 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, is the sequence whose nth term is .
(2). Defining a sequence by a formula for its nth term.
For example, the sequence can also be written as or or simply .
(3). Defining a sequence by a Recursion formula, that is, by a rule which expresses the nth term in terms of the term.
For example, let , for all .
These relations define a sequence whose nth term is .
Examples of Sequence:
- or , for all is a sequence.
- is a sequence .
- is a sequence .
- is a sequence .
- is a sequence .
- is a sequence .
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