# Introduction to Sequences

# Sequences:

In Mathematics, We define Sequences as –

A mapping (s) of Natural Numbers (N) to some Non empty set, (say-A)

s : N → A

Further we discuss the types of Sequences, Range, Methods of writing Operations on them along with Some theorems related to Sequences..

## Methods of Writing Sequences:

There are three methods of writing a Sequence; Such as-

- Listing Or Roster Method
- Set builder Or Rule Method
- Inductive Or Recursive Method

## Listing Or Roster Method-

In this method, we list all terms in an angular bracket. such as-

Example: s=<1,3,5,7,…> , s= Seq. of odd numbers.

## Set builder Or Rule Method–

In this method, Seq. is specified by a rule or formula for the term s_{n} in the respective Seq.

Example: s=<2n-1|n∈N>, where s is a seq of odd numbers.

## Inductive Or Recursive Method-

In this method,

- s
_{1}is given - We have to find s
_{n+1}, n=1 using a formula in terms of s_{n}or s_{1},s_{2},…

Example: s_{1}=1, s_{n+1}= s_{n}+2 for n=1.

## Subsequence:

Let <s_{n}> be a sequence & let r_{1}<r_{2}<… be a strictly increasing sequence of natural numbers. Then –

s’= <s_{r1},s_{r2},…> is a subsequence of s.

# Types of Sequences:

## Constant Sequence-

A Sequence of the form X=<x,x,…> all of whose terms equal to a constant x?R.

Example: a sequence of terms (1)^{n}=<1,1,1…>

## Bounded Sequence-

A Sequence is bounded if it has some real number which binds it. Conversely, A Sequence is unbounded if it has no real number which binds it.

# Range of Sequence-

The Range of sequence s : N ∈ A is a set

R (s) = { s(1) ,s(2) ,s(3) . . . }

Basically, Range is a set of the elements in a sequence. The elements of range are ‘terms’.

# Operations on Sequences-

Let us consider sequences s=<s_{n}>, t=<t_{n}> & c∈R. We define them as follows-

- Sum : s+t=<s
_{n}+t_{n }| n∈N> - Difference :s-t=<s
_{n}– t_{n }| n∈N> - Scalar Multiplication: cs=<cs
_{n}| n∈N> - Product: st=<s
_{n}t_{n }| n∈N> - Division: s/t=<s
_{n}/ t_{n }| t_{n }& n∈N>

Above all is about the basic introduction of Sequences. There is lot more about it. Get it in next posts.