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Introduction to Sequence and Series

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Sequence: The set of numbers arranged in definite pattern is called sequence.
Example: 2,4,6,8,10,12……

Series: If the set of numbers arranged in definite pattern are 
connected either by plus or minus sign, then it is called series.
Example: 2+4+6+8+10+12+………


Types of Sequence:

Arithmetic Sequence
→The sequence whose common difference between the two successive terms are same is called arithmetic sequence.
For example:3,7,11,15........
Here,
The common difference between the two successive terms is same i.e.
7-3 =4, 11-7=4, 15-11 =4.
Here, The common difference between two successive terms is 4.
So, the given sequence is arithmetic sequence. The difference could be positive, negative or even in fraction.


Geometric Sequence:
→The sequence whose common ratio between the two successive terms are same is called geometric sequence.
For example:3, 9, 27, 81……..
Here,
T
he common ratio between the two successive terms is same i.e.,
81/27=3, 27/9=3, 9/3=3
Here,
The common ratio between two successive terms is 3.
So, the given sequence is geometric sequence.
The ratio could be positive, negative or even in fraction.


Harmonic Sequence(not imp for class 10 pov but imp for class 11)
→The sequence formed by the reciprocals of terms of  arithmetic sequence is called harmonic sequence.
For example: 1/2, 1/4, 1/6, 1/8.........
Here, 
The reciprocals of the terms in order are in arithmetic sequence i.e.
2, 4, 6, 8..........
Here, 
The reciprocals have common difference of 2.
So, the given sequence is harmonic sequence.