Introduction to Sequences
The word sequence denotes certain objects or events occurring in the same order. In itself, a sequence is a set of numbers arranged in a specific manner. The applications of sequences can be found in many areas including in the process of analyzing economic data and certain areas of physics. Also, sequences are important in developing mathematical topics which themselves have wide technical applications.
Sequences are also called progressions, and they are used to represents ordered lists of numbers. As the members of a sequence are in a definite order, a correspondence can be established by matching them one by one with the numbers $$1,2,3,4, \ldots $$ For example, if the sequence is $$1,4,7,10, \ldots $$ nth member, then such a correspondence can be set up as shown in the diagram below:
\[\begin{array}{*{20}{c}} {{\text{Position}}}&{}&{{\text{Member of Sequence}}} \\ {\text{1}}& \to &{\text{1}} \\ {\text{2}}& \to &4 \\ 3& \to &7 \\ 4& \to &{10} \\ \vdots & \vdots & \vdots \\ n& \to &{n{\text{th member}}} \end{array}\]
Thus a sequence is a function whose domain is a subset of the set of natural numbers. A sequence is a special type of function from a subset of $$\mathbb{N}$$to $$\mathbb{R}$$or$$\mathbb{C}$$. Sometimes the domain of sequence is taken to be a subset of the set $$\left\{ {0,1,2,3, \ldots } \right\}$$, i.e., the set of non-negative integers. If all members of a sequence are real numbers, then it is called a real sequence.
Sequences are usually named with letters, $$a,b,c$$ etc., and $$n$$ is used instead of $$x$$ as a variable. If a natural number $$n$$ belongs to a domain of a sequence $$a$$, the corresponding elements in its range are denoted by $${a_n}$$. For convenience, a special notation $${a_n}$$ is adopted for $$a\left( n \right)$$ and the symbol $$\left\{ {{a_n}} \right\}$$ or $${a_1},{a_2},{a_3}, \ldots ,{a_n}, \ldots $$ is used to represent the sequence $$a$$. The elements in the range of the sequence $$a$$ are called its terms; that is, $${a_1}$$ is the first term, $${a_2}$$ the second term and $${a_n}$$ the nth term or the general term.
For example, the terms of the sequence $$\left\{ {n + {{\left( { – 1} \right)}^n}} \right\}$$ can be written by assigning to $$n$$ the values $$1,2,3, \ldots $$. If we denote the sequence by $$\left\{ {{b_n}} \right\}$$, then $${b_n} = n + {\left( { – 1} \right)^n}$$ and we have:
\[\begin{array}{*{20}{c}} {{b_1}}& = &{1 + {{\left( { – 1} \right)}^1}}& = &{1 – 1}& = &0 \\ {{b_2}}& = &{2 + {{\left( { – 1} \right)}^2}}& = &{2 + 1}& = &3 \\ {{b_3}}& = &{3 + {{\left( { – 1} \right)}^3}}& = &{3 – 1}& = &2 \\ {{b_4}}& = &{4 + {{\left( { – 1} \right)}^4}}& = &{4 + 1}& = &5 \end{array}\]
If the domain of a sequence is a finite set then the sequence is called a finite sequence; otherwise it is called an infinite sequence.
Note: An infinite sequence has no last term.
Some examples of sequences are:
- $$1,4,9, \ldots ,121$$
- $$1,3,5,7,9, \ldots ,21$$
- $$1,2,4, \ldots $$
- $$1,3,7,15,31, \ldots $$
- $$1,6,20,56, \ldots $$
- $$1,\frac{1}{3},\frac{1}{5},\frac{1}{7},\frac{1}{9}, \ldots $$
Sequences (1) and (2) are finite whereas sequences (3) to (6) are infinite.