Sequence and Series
Sequence and series are used in mathematics as well as in our daily lives. A sequence is also known as progression and a series is developed by sequence. Sequence and series is one of the basic concepts in Arithmetic. Sequences are the grouped arrangement of numbers orderly and according to some specific rules, whereas a series is the sum of the elements in the sequence. For example, 2, 4, 6, 8 is a sequence with four elements and the corresponding series will be 2 + 4 + 6+ 8, where the sum of the series or value of the series will be 20.
There are various types of sequences and series depending upon the set of rules that are used to form the sequence and series. Sequence and series are explained in detail below.
1.  What Are Sequence and Series? 
2.  Difference Between Sequence and Series 
3.  Types of Sequence and Series 
4.  Sequence and Series Formulas 
5.  FAQs on Sequence and Series 
What Are Sequence and Series?
The sequence is the group or sequential arrangement of numbers in a particular order or set of rules. Series is formed by adding the digits of a sequence. In a sequence, an individual term can be present in many places. Sequences can be of two types, i.e. infinite terms sequence and finite terms sequence and series will be then defined by adding the terms of the sequence. Sum of infinite terms in a series is possible in some cases as well.
Let us understand this with an example. 1, 3, 5, 7, 9, 11, ... is a sequence where there is a common difference of 2 between any two terms and the sequence goes on increasing up to infinity unless the upper limit is given. These types of sequences are known as arithmetic sequences. Now if we add the numbers in the sequence like 1 + 3 + 5 + 7+ 9... this will make a series of this sequence. These kinds of series are known as arithmetic series. Few examples of sequence and series are given in the image shown below:
Difference Between Sequence and Series
The important differences between sequence and series are explained in the table given below:
Sequence  Series 

In sequence, elements are placed in a particular order following a particular set of rules.  In series, the order of the elements is not necessary. 
A definite pattern of the numbers is important.  The pattern of the numbers is not important. 
Order of appearance of the numbers is important.  The order of appearance is not important. 
Example: Harmonic sequence: 1, 1/2, 1/3, 1/4, 1/5, 1/6...  Example: Fourier series: f(x) = 4h/π ( sin(x) + sin(3x)/3 + sin(5x)/5 + ... ) 
Types of Sequence and Series
There are various types of sequences and series, in this section, we will discuss some special and most commonly used sequences and series. The types of sequence and series are:
 Arithmetic Sequences and Series
 Geometric Sequences and Series
 Harmonic Sequences and Series
Arithmetic Sequence and Series
An arithmetic sequence is a sequence where the successive terms are either the addition or subtraction of the common term known as common difference. For example, 1, 4, 7, 10, ...is an arithmetic sequence. A series formed by using an arithmetic sequence is known as the arithmetic series for example 1 + 4 + 7 + 10... is an arithmetic series.
Geometric Sequence and Series
A geometric sequence is a sequence where the successive terms have a common ratio. For example, 1, 4, 16, 64, ...is an arithmetic sequence. A series formed by using geometric sequence is known as the geometric series for example 1 + 4 + 16 + 64... is a geometric series. The geometric progression can be of two types: Finite geometric progression and infinite geometric series.
Harmonic Sequence and Series
A harmonic sequence is a sequence where the sequence is formed by taking the reciprocal of each term of an arithmetic sequence. For example, 1, 1/4, 1/7, 1/10,... is a harmonic sequence. A series formed by using harmonic sequence is known as the harmonic series for example 1 + 1/4 + 1/7 + 1/10.... is a harmonic series.
Sequence and Series Formulas
There are various formulas related to various sequences and series by using them we can find a set of unknown values like the first term, nth term, common parameters, etc. These formulas are different for each kind of sequence and series. Formulas related to various sequences and series are explained below:
Arithmetic Sequence and Series Formula
The various formulas used in arithmetic sequence are given below:
Arithmetic sequence  a, a + d, a + 2d, a + 3d, ... 
Arithmetic series  a + (a + d) + (a + 2d) + (a + 3d) + ... 
First term:  a 
Common difference(d):  Successive term – Preceding term or \(a_n  a_{n1}\) 
nth term\(a_n\)  a + (n1)d 
Sum of arithmetic series \(S_n\)  (n/2)(2a + (n1)d) 
Geometric Sequence and Series Formulas
The various formulas used in geometric sequence are given below:
Geometric sequence  a, ar, ar^{2},….,ar^{(n1)},… 
Geometric series  a + ar + ar^{2 }+ ...+ ar^{(n1)}+ … 
First term  a 
Common ratio  r 
nth term  ar^{(n1)} 
Sum of geometric series 
Finite series: \(S_n\) = a(1−r^{n})/(1−r) for r≠1, and \(S_n\) = an for r = 1 Infinite series: \(S_n\) = a/(1−r) for r < 1, and not defined for r > 1 
Sequence and Series Tips
The following points are helpful to clearly understand the concepts of sequence and series.
 In an arithmetic sequence and series, a is represented as the first term, d is a common difference, \(a_n\) as the nth term, and n as the number of terms.
 In general, the arithmetic sequence can be represented as a, a+d, a+2d, a+3d,...
 Each successive term is obtained in a geometric progression by multiplying the common ratio to its preceding term.
 The formula for the nth term of a geometric progression whose first term is a and common ratio is r is \(a_n\) = ar^{n−1}
 The sum of infinite GP formula is given as: \(S_n\) = a/(1−r) where r<1.
Related Topics on Sequence and Series
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Sequence and Series Examples

Example 1: What will be the 15th term of the arithmetic sequence 3, (1/2), 2…. using sequence and series formula?
Solution: Given a = 3, d = (1/2) (3) = 5/2, n = 15
Using the formula for nth term of an arithmetic sequence and series:
\(a_n\) = a+(n1)dPutting the known values:
\(a_{15}\) = 3 +(151) 5/2
\(a_{15}\) = 32
Answer: The 15th term of the given arithmetic sequence is 32.

Example 2: Find the next term of the given geometric sequence: 1, 1/2, 1/4, 1/8 ... using sequence and series formula
Solution:
Given: a = 1, r = (1/2)/1 = 1/2
To find: 5th term
Using the formula for the nth term of a geometric sequence and series:
\(a_n\) = ar^{(n1)}Putting the known values in the formula:
\(a_5\) = 1(1/2)^{(51)}
\(a_5\) = (1/2)^{(4)}
\(a_5\) = 1/16
Answer: The next term of the sequence is 1/16.
FAQs on Sequence and Series
What are Sequence and Series?
Sequence and series are used in mathematics as well as in our daily lives. The sequence is the group or sequential arrangement of numbers in a particular order or set of rules. Series is formed by adding the digits of a sequence.
What is the Difference Between Sequence and Series?
In sequence, elements are placed in a particular order following a particular set of rules, a definite pattern of the numbers is important, and order of appearance of the numbers is important. In series, the order of the elements is not necessary, the pattern of the numbers is not important, and the order of appearance is not important. Example of sequence: Harmonic sequence: 1, 1/2, 1/3, 1/4, 1/5, 1/6... . Example of series: Fourier series: f(x) = 4h/π ( sin(x) + sin(3x)/3 + sin(5x)/5 + ... )
What are Arithmetic Sequence and Series?
An arithmetic sequence is a sequence where the successive terms are either the addition or subtraction of the common term known as common difference. For example, 1, 4, 7, 10, ...is an arithmetic sequence. A series formed by using an arithmetic sequence is known as the arithmetic series for example 1 + 4 + 7 + 10... is the arithmetic series.
What is Geometric Sequence and Series?
A geometric sequence is a sequence where the successive terms have a common ratio. For example, 1, 4, 16, 64, ...is an arithmetic sequence. A series formed by using geometric sequence is known as the geometric series for example 1 + 4 + 16 + 64... is a geometric series. The geometric progression can be of two types: Finite geometric progression and infinite geometric progression.
What is the Similarity Between Sequence and Series?
The sequence and the series of the same type, both are made up of the same elements. A series is formed by using the elements of the sequence and joining them by the addition symbol.
What are the Different Types of Sequence?
Sequences are can be of various types. We see sequences in almost every other situation. The most commonly observed and used sequences are:
 Arithmetic Progression(AP)
 Geometric Progression (GP)
 Harmonic Progression (HP)
What is Harmonic Sequence and Series?
A harmonic sequence is a sequence where the sequence is formed by taking the reciprocal of each term of an arithmetic sequence. For example, 1, 1/4, 1/7, 1/10,... is a harmonic sequence. A series formed by using harmonic sequence is known as the harmonic series for example 1 + 1/4 + 1/7 + 1/10.... is a harmonic series.
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