Flashcards: Mathematical Induction | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


 
 
 
FIRST PRINCIPLE OF MATHEMATICAL 
INDUCTION 
The proposition ?? ( ?? ) involving a natural number ?? is assumed to be true for all ?? ? ?? , follows the 
following three steps : 
Step - I (Verification step) 
Actual verification of the proposition ?? ( ?? ) for the starting value of ?? = 1 
Step - II (Induction step) 
Assuming that if ?? ( ?? ) is true for ?? = ?? ; ?? = 1, prove that it is also true for ?? = ?? + 1. 
Step - III (Generalization step) 
Combining the above two steps leads to the conclusion that ?? ( ?? ) is true for all integers ?? ? ?? . 
  
Page 2


 
 
 
FIRST PRINCIPLE OF MATHEMATICAL 
INDUCTION 
The proposition ?? ( ?? ) involving a natural number ?? is assumed to be true for all ?? ? ?? , follows the 
following three steps : 
Step - I (Verification step) 
Actual verification of the proposition ?? ( ?? ) for the starting value of ?? = 1 
Step - II (Induction step) 
Assuming that if ?? ( ?? ) is true for ?? = ?? ; ?? = 1, prove that it is also true for ?? = ?? + 1. 
Step - III (Generalization step) 
Combining the above two steps leads to the conclusion that ?? ( ?? ) is true for all integers ?? ? ?? . 
  
 
 
 
SECOND PRINCIPLE OF MATHEMATICAL 
INDUCTION (Extended Principle) 
Sometimes, the first principle of mathematical induction does not suffice. In such cases we use the 
extended principle as below: 
Step - I (Verification step) 
We verify that ?? ( ?? ) is true for 1 and 2 both. 
Step - II (Induction step) 
Assume that ?? ( ?? ) is true for ?? = ?? and ?? = ?? + 1 , ?? = ?? , prove that ?? ( ?? ) is true for ?? = ( ?? + 2 ). Step - III 
(Generalization step) 
Combining the above two steps leads to the conclusion that ?? ( ?? ) is thus true ? ?? ? ?? . 
  
Page 3


 
 
 
FIRST PRINCIPLE OF MATHEMATICAL 
INDUCTION 
The proposition ?? ( ?? ) involving a natural number ?? is assumed to be true for all ?? ? ?? , follows the 
following three steps : 
Step - I (Verification step) 
Actual verification of the proposition ?? ( ?? ) for the starting value of ?? = 1 
Step - II (Induction step) 
Assuming that if ?? ( ?? ) is true for ?? = ?? ; ?? = 1, prove that it is also true for ?? = ?? + 1. 
Step - III (Generalization step) 
Combining the above two steps leads to the conclusion that ?? ( ?? ) is true for all integers ?? ? ?? . 
  
 
 
 
SECOND PRINCIPLE OF MATHEMATICAL 
INDUCTION (Extended Principle) 
Sometimes, the first principle of mathematical induction does not suffice. In such cases we use the 
extended principle as below: 
Step - I (Verification step) 
We verify that ?? ( ?? ) is true for 1 and 2 both. 
Step - II (Induction step) 
Assume that ?? ( ?? ) is true for ?? = ?? and ?? = ?? + 1 , ?? = ?? , prove that ?? ( ?? ) is true for ?? = ( ?? + 2 ). Step - III 
(Generalization step) 
Combining the above two steps leads to the conclusion that ?? ( ?? ) is thus true ? ?? ? ?? . 
  
 
 
 
 
 
Note: The second principle of mathematical induction is useful to prove recurrence relations which 
involve three successive terms e.g., ?? ?? ?? + 1
= ?? ?? ?? + ?? ?? ?? - 1
. 
  
Page 4


 
 
 
FIRST PRINCIPLE OF MATHEMATICAL 
INDUCTION 
The proposition ?? ( ?? ) involving a natural number ?? is assumed to be true for all ?? ? ?? , follows the 
following three steps : 
Step - I (Verification step) 
Actual verification of the proposition ?? ( ?? ) for the starting value of ?? = 1 
Step - II (Induction step) 
Assuming that if ?? ( ?? ) is true for ?? = ?? ; ?? = 1, prove that it is also true for ?? = ?? + 1. 
Step - III (Generalization step) 
Combining the above two steps leads to the conclusion that ?? ( ?? ) is true for all integers ?? ? ?? . 
  
 
 
 
SECOND PRINCIPLE OF MATHEMATICAL 
INDUCTION (Extended Principle) 
Sometimes, the first principle of mathematical induction does not suffice. In such cases we use the 
extended principle as below: 
Step - I (Verification step) 
We verify that ?? ( ?? ) is true for 1 and 2 both. 
Step - II (Induction step) 
Assume that ?? ( ?? ) is true for ?? = ?? and ?? = ?? + 1 , ?? = ?? , prove that ?? ( ?? ) is true for ?? = ( ?? + 2 ). Step - III 
(Generalization step) 
Combining the above two steps leads to the conclusion that ?? ( ?? ) is thus true ? ?? ? ?? . 
  
 
 
 
 
 
Note: The second principle of mathematical induction is useful to prove recurrence relations which 
involve three successive terms e.g., ?? ?? ?? + 1
= ?? ?? ?? + ?? ?? ?? - 1
. 
  
 
 
APPLICATION OF MATHEMATICAL 
INDUCTION 
I. Identities Type Problems 
II. Divisibility Type Problems 
To prove that ?? ( ?? ) is divisible by ?? , following the procedure is followed 
(I) First we show that ?? ( 1 ) is divisible by ?? . 
(II) Assuming that ?? ( ?? ) is divisible by ?? , it is proved that ?? ( ?? + 1 ) is also divisible by ?? . For this either 
divide ?? ( ?? + 1 ) with ?? ( ?? ) and show that remainder is divisible by ?? . 
or 
Split ?? ( ?? + 1 ) = ?? ( ?? ) . ?? + ?? ; ?? ? ?? and show that ?? is divisible by ?? . 
III. Inequalities Type Problems 
IV. Problems Based on Extended Principle of Mathematical Induction If the given problem cannot be 
solved by direct use of principle mathematical induction, try to use the extended principle of 
mathematical induction.