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Sequence and series are two different concepts that are used in mathematics for different purposes. But what is sequence? Sequences are patterns or order of certain elements in a balanced manner whereas a series is the sum of that sequence or order. A sequence of elements tells about the type of progression that is followed in that pattern. Series is a topic more involved in Differential equations. There are various types of sequences and series which have different methods to solve the equations. This article focuses on the key features of series and sequence and the difference between them.
What is a sequence?
Definition of sequence: It is the pattern in which the elements of a progression are arranged. The relation between the preceding and succeeding terms of the sequence should be equal.
For e.g. if there is a sequence like
2,5,8,11,14……….
If we notice the above sequence carefully, it follows a pattern where the difference between every term is 3. Therefore 3 is a common difference.
The sequence could be finite or infinite. In the above example, the ….. After 14, it shows that the sequence is infinite.
There are some types of sequences like:
Arithmetic sequence
The general expression for an arithmetic sequence looks like this:
a,a+d,a+2d,a+3d,a+4d……….
where a is the first term and d is a common difference.
An example of an arithmetic progression is 2,6,10,14,18….
Here a=2,d=4
Geometric sequence
The general expression for a geometric sequence looks like this:
a,ar,ar2,ar3……..
Where a is the first term and r is the common ratio
An example of a geometric progression is 3,6,12,24…..
Here a=3,r=2
Harmonic sequence
The general expression for a harmonic sequence looks like this:
1/a,1/a+d,1/a+2d,1/a+3d……
The reciprocal of all the entities are in an arithmetic sequence.
An example of harmonic progression is ¼,⅙,⅛,1/10…..
Fibonacci sequence
In this sequence, each term is obtained by adding the preceding two numbers.
Example of a fibonacci sequence is 0,1,1,2,3….
Series
The sum of the sequence is called a series. It is the addition of the elements of the sequence. There are many formula of series. There are some types of series like:
Arithmetic series
The general form of this series is: a+(a+d)+(a+2d)+(a+3d)+…..
Where a=first term, d=common difference of the sequence.
For eg. 5+7+9+11…….
Geometric Series
The general form of this series is: a + ar2 + ar3 + ar4 +……..
Where a=first term, r=common ratio of the sequence.
For eg. 4+8+16+32…….
Harmonic Series
The general form of this series is: 1/a + 1/a+2d + 1/a+3d + 1/a+4d…..
For eg. ⅓ + ⅙ + 1/9 +………
Alternating Series
Power Series
Exponent Series
Difference between Sequence and Series
Sequence |
Series
|
In a sequence, the elements are arranged in a symmetrical pattern, thus following a particular order. | Series is the summation of the sequence of the elements of that pattern which can be understood only after we know what is sequence. |
An example of a sequence would be 2,5,8,11…. | An example of a series would be 2+5+8+11…. |
In a sequence, the order of the sequence is important | In series, the order doesn’t really matter since it is just the summation of the terms. |
Some types of sequences are arithmetic sequence, geometric sequence, fibonacci sequence, etc. | Some types of series are arithmetic series, exponential series, harmonic series, etc. |
All the entities in this arrangement follow a particular pattern, and it is the same with the succeeding as well as the preceding element. | It is just the sum of the sequence which is arranged in a pattern. |
The order of a sequence is important and is not the same if reversed. For e.g. 2,5,8 is not the same as 8,5,2 | In series, the order is not important at all and the answer remains the same even if the order is changed or reversed. For eg. 2+5+8 is same as 8+5+2 |
Many examinations like JEE and State Board examinations ask sequence and series problems in their exam.
Conclusion
Sequences are series, though very similar topics, but still are very different from each other. They share the same concept but the key features are very different from each other. The order of the pattern is very important in a sequence but does not change anything in a series. Series is derived from the sequence. Series is nothing but the sum of all those elements present in the sequence. There are various sequence and series problems to which we apply formulae after we know what is sequence that is present in the question, which makes the solving process easier and efficient and less time consuming, thus proving to be quite beneficial.
