Informally, if we want to define sequence (seq) it will be something as an arrangement of events, elements, terms etc. It is a manner or discipline to keep the things in order. It may be a set of members. The length of the same is determined by the number of ordered elements. It is the arrangement of similar objects. The same elements can appear many times at different positions in the same arrangement.
A function f(x) which has domain and range, where x may be set of the natural numbers is called as its definition. The seq are of the following types.
( 1) Finite
Finite seq:- In finite form the number of elements is countable. For example
A= ( 1,3,5,7,……………111)
Seq of any finite length ‘n is termed as an n-tuple and Finite seqs might also include empty form ( ) which will have no elements.
Infinite seq:- In the Infinite form the number of elements is not countable. For example
A =(……….. -3,-2,-1,0,1,2,3……………….)
Infinite seq is infinite in both directions. It has neither a first nor a final element is called a bi-infinite or two-way infinite. For example a function from all the integers included into a set, such that the seq of all even integers ( …-8,-6 -4, -2, 0, 2, 4, 6, 8,10,12… ), is found to be bi-infinite.
There are many important integer sequential forms and these are as follows.
(A) The even numbers which can be divided by 2.
(B) The odd numbers which cannot be divided by 2.
(C) The prime numbers that have no divisors except 1 and themselves.
The Fibonacci number Sequences:- It is nothing but in which elements are the sum of the previous two elements. The first two elements are either 0 or 1. This is (0,1,1,2,3,5,8,13,21,34,65,99...).
Formula for the Fibonacci:- It can be defined using a recursive rule along with two initial elements.
, with a0 = 0 and a1 = 1.
Where, 0 and 1 are initial elements of the Fibonacci sequence.
Special seq:- some of the special seq forms are given below.
(3) Square of numbers
(5) cube of numbers
(6) Roots of numbers
(7) cubic roots of numbers
(8) A set of vowels.
(9) indexing of the documents
Examples and notation
It is a list of elements with a particular order. These are useful for the study of the functions, spaces, and other mathematical structures by using the properties of convergence . The basis for series is sequences. These are used in differential equations and analysis. These are also used to find the patterns or to solve the puzzles and can be used in the study of prime numbers.