Types of Sequence and Examples

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 sequence 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_na_{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

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

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