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 a₁, a₂, a₃,…, 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, a₁ is the first term, a₂ 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

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

a_n=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

 

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