**What is Sequence?**

Sequences are also called progression. Sequences are used to represent ordered lists of members. As the member of a sequence is indefinite order so correspondence can be established by matching them one by one with the numbers 1, 2, 3, 4,…

**Example of Sequence**

for example, if the sequence is 1, 4, 7, 10, … . nth member then such a correspondence can be set up as follows

thus a sequence is a function whose domain is a subsequence of the set of natural numbers. A sequence is a special kind of function from a subset of n to R or C. Sometimes the domain of a sequence is taken to be a subset of the set { 0, 1, 2, 3,…}, i.e. the set of nonnegative integers. If all the members of a sequence are real then it is called a **real sequence.**

**Notation for Sequence**

Sequences are usually denoted with letters a, b, c, etc., and n is used rather than x as a variable. If the natural number n be a part of the domain of a sequence a, the reciprocal element in its range is denoted by an. A special notation an is adopted for a(n) and the symbol { a_n } or a_1 , a_2, a_3 …, a_n ,… is used to represent the sequence.

The elements in the range of a sequrence{ a_n } are called its **terms** that is a_1 is the first term, a_2 is the second term, and a_n is the nth term or general term of the sequence.

for example, the term of the sequence {n+(-1)^n} can be written by assigning to n, the values 1, 2, 3, … if we denote the sequence by { b_n }, then

b_n = n+ (-1)^n and we have

b_1 = 1+ (-1)^1 = 1-1 =0

b_2 = 2+ (-1)^2 = 2+1 =3

b_3 = 3+ (-1)^3 = 3-1 =2

b_4 = 4+ (-1)^4 =4+1 =5etc

**Note for Sequence:**

If the domain of the sequence is a finite set, then it is called the **finite sequenc**e otherwise an** infinite sequence**.

An infinite sequence has no last term.

**Types of sequences **

**Arithmetic sequence****Geometric sequence****Harmonic sequence****Fibonacci sequence**

**Define Arithmetic sequence**

A sequence { a_n } is an arithmetic sequence if a_n – a_{n-1} is the same for all n belongs to N and n>1.

In other words, the difference of two consecutive terms of a sequence is the same then it is called an arithmetic sequence. The difference is called the common difference. The arithmetic sequence is also called the arithmetic progression. The d is the notation that is used for denoting the difference between two terms.

**Example** **of Arithmetic sequence**

5, 10, 15, 20, 25,….

Here the common difference is 5

If in the arithmetic progression the first term is a_1 and the common difference is d then the nth term of this sequence is given

a_n = a_1 +(n-1)d this is also called the general term.

**What is Geometric sequence**?

A sequence { a_n } is said to be a geometric sequence if a_n \ a_{n-1} is the same non-zero number for all n belongs to N and n>1.

In other words, if the successive terms of a sequence have the same ratio between them then it is called a geometric sequence. the other name of a geometric sequence is a geometric progression. The term a_n \ a_{n-1} is the common ratio and r is the notation of this term. It is clearly defined that r is the ratio of any term of the G.P to its predecessor. There is no term of a geometric sequence that is zero.

**Example ****Geometric sequence**

**Geometric sequence**

1, 3, 9, 27,…

the nth term of a geometric sequence is given as

a_n = a_1r^{n-1}

**What is Harmonic sequence**?

A sequence is called the harmonic sequence if the reciprocals of its term are in an arithmetic sequence. It is also known as harmonic progression. Since the reciprocal of 0 is not defined so zero is not the term of harmonic sequence

**Example** **Harmonic sequence**

1,\frac13, \frac15 , \frac17

This is a harmonic sequence and their reciprocal 1 3 5 7 are in a geometric sequence.

If a_1 is the first term of harmonic sequence and d is the difference then the nth term is given as

1\ a_1 +(n-1)d

**Define Fibonacci sequence**?

In any sequence, we are adding the two-term to get the next term this process is continued until we get our required term this type of sequence is called the Fibonacci sequence.

**Example Define Fibonacci sequence**

3, 1, 4, 5, 9,14,…

In this sequence, we can see that

a_1=3 and a_2= 1

a_3 = a_2+ a_1 = 3+ 1 =4

a_4 = a_3 + a_2 = 4 + 1 =5 and so on.

So the Fibonacci Sequence for the nth term is given as

a_n = a_{n-2}+ a_{n-1}, n>2

There are some other types of sequences that are we discuss here as An **integer sequence** is a sequence in which elements are integers. A **polynomial sequence** is a sequence in which elements are polynomial. A **binary sequence** is a sequence in which elements have one of two distinct values like that a term coin which has only two values head or tail.

**Increasing and Decreasing of Fibonacci **

A sequence is **monotonically increasing** if each term in the sequence is greater or equal to the previous term.

**Example** **of Fibonacci**

1, 4, 7, 9,

A sequence is **monotonically decreasing** if each term of the sequence is less than or equal to the previous term

**Example** **of Fibonacci**

10, 6, 6, 4