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-

  1. Listing Or Roster Method
  2. Set builder Or Rule Method
  3. 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 sn 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,

  1. s1 is given
  2. We have to find sn+1, n=1 using a formula in terms of sn or s1,s2,…

Example: s1=1, sn+1= sn+2 for n=1.

Subsequence:

Let <sn> be a sequence & let r1<r2<… be a strictly increasing sequence of natural numbers. Then –

s’= <sr1,sr2,…> 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=<sn>, t=<tn> & c∈R. We define them as follows-

  1. Sum : s+t=<sn+t| n∈N>
  2. Difference :s-t=<sn– t| n∈N>
  3. Scalar Multiplication: cs=<csn| n∈N>
  4. Product: st=<snt| n∈N>
  5. Division: s/t=<sn/ t| tn & n∈N>

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