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# Sequence-Finite Sequence and Infinite Sequence

July 14, 2022 written by Azhar Ejaz

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 an.

For convenience, a special notation an is adopted for a(n) and the symbol {an} or a1, a2, a3,…, an is used to represent the sequence a.

The elements in the range of the sequence {an} are called its terms; that is, a1 is the first term, a2 the second term, and an 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 {bn}. then

bn = n + (-1)n and we have

n=1

b1 = 1 + (-1)1 = 1 -1 = 0

n=2

b2 = 2 + (-1)2 = 2 +1 = 3

n=3

b3 = 3 + (-1)3 = 3 -1 = 2

n=4

b4 = 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

an=2n-3

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

n=1

a1=2(1)-3

a1=2-3

a1=-1

n=2

a2=2(2)-3

a2=4-3

a2=1

n=3

a3=2(3)-3

a3=6-3

a3=3

n=4

a4=2(4)-3

a4=8-3

a4=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:

a1=2

a2=a1+4=2+4=6

a3=a2+5=6+5=11

a4=a3+6=11+6=17

a5=a4+7=17+7=24

a6=a5+8=24+8=32

a7=a6+9=32+9=41

2) 1,3,12,60,….a6

Solution: a1=1

a2=a1x3=1×3=3

a3=a2x4=3×4=12

a4=a3x5=12×5=60

a5=a4x6=60×6=360

a6=a5x7=360×7=2520

### 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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