# Sequence-Finite Sequence and Infinite Sequence

A sequence is an ordered set of** numbers** formed according to some definite rule. As the members of a sequence are in a definite order, correspondence can be established by matching them one by one with the numbers 1, 2, 3, and 4, ….

**For Example:**

If the sequence is 1, 4, 7, 10, …, nth member, then such a correspondence can be set up as shown below

**Position The member of the sequence**

** ** 1 ———————–> 1

2————————> 4

** **3 ————————>7

4 ———————–>** **10** **

** . .**

** . .**

** **n ————————> Nth member

** **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 N to R or C.

## Real Sequence

*If all members of a sequence are* real numbers* then it is called a real sequence.*

## Complex sequence:

*Thus a sequence is a function whose range is a subset of *** complex number**

*than is called a complex sequence.*

Sequences are usually named with the letters a, b, and c,.. and n is used instead of x as a variable. If a natural number n belongs to the domain of a sequence a, the corresponding element in its range is denoted by a_{n}.

For convenience, a special notation a_{n} 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 a.

The elements in the range of the sequence {a_{n}} 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.

**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

n=1

b_{1} = 1 + (-1)^{1} = 1 -1 = 0

n=2

b_{2} = 2 + (-1)^{2} = 2 +1 = 3

n=3

b_{3} = 3 + (-1)^{3} = 3 -1 = 2

n=4

b_{4} = 4 + (-1)^{4} = 4 +1 = 5 etc

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

## Finite Sequence

*If the domain of a sequence is finite then the sequence is finite.*

** For example:**

(1) 1, 4, 9, …. 12

(2) 1, 3, 5, …21

## Infinite Sequence:

*If the domain of a sequence is unlimited then the sequence is called an infinite sequence.*

An infinite sequence has no last term

**For example:**

(1) 1, 3, 7, …….

(2) 1, 6, 20, …

**For example:**

Write the first four terms of the following sequence if

**a _{n}=2n-3**

Solution: put n=1,2,3,4

n=1

a_{1}=2(1)-3

a_{1}=2-3

a_{1}=-1

n=2

a_{2}=2(2)-3

a_{2}=4-3

a_{2}=1

n=3

a_{3}=2(3)-3

a_{3}=6-3

a_{3}=3

n=4

a_{4}=2(4)-3

a_{4}=8-3

a_{4}=5

The first four terms are

-1,1,3,4

**For example:**

Find the indicated terms of the following sequence.

**2,6,11,17,…….a7**

**Solution:**

a_{1}=2

a_{2}=a_{1}+4=2+4=6

a_{3}=a_{2}+5=6+5=11

a_{4}=a_{3}+6=11+6=17

a_{5}=a_{4}+7=17+7=24

a_{6}=a_{5}+8=24+8=32

a_{7}=a_{6}+9=32+9=41

**2) 1,3,12,60,….a _{6}**

Solution: a1=1

a_{2}=a_{1}x3=1×3=3

a_{3}=a_{2}x4=3×4=12

a_{4}=a_{3}x5=12×5=60

a_{5}=a_{4}x6=60×6=360

a_{6}=a_{5}x7=360×7=2520

## Frequently Asked Question-FAQs

### What is sequence and example?

A sequence is an ordered, numbered list where the three dots (…) mean to continue forward in the pattern established. Each number in the sequence is referred to as a term; for example, in the sequence 1, 3, 5, 7, 9 … 1 is the first term, 3 is the second term, 5 is the third term, and so on.

### what is Real Sequence?

If all members of a sequence are real numbers then it is called a real sequence

### what is Complex sequence?

Thus a sequence is a function whose range is a subset of complex numbers than is called a complex sequence*.*

### What is finite sequence example?

A finite sequence is a sequence that contains only a finite number of terms – for example, 1, 2, 3, 4, 5.

### What is infinite sequence?

If the domain of a sequence is unlimited then the sequence is called an infinite sequence.

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