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Class 11 Revision Notes Principle of Mathematical Induction

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Mathematics – Iii

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Class XI Maths 1

Revision Notes

Class 11 Maths

Chapter 4 - The Principle of Mathematical Induction

1. Example: Rohit is a man, and all men consume food, hence Rohit eats food: Generalization of Specific Instance

2. Rohit, for example, eats. Vikash consumes food. Vikash and Rohit are bothInduction: Specific Instances to Generalization

males. The statement All men consume food is true forn = k + 1 , and it is also true for all natural integers n. n = 1 , n =k , and

3 of Principle of Mathematical Induction: Allow P n

( )to be a result or statement expressed in terms of n. (given question).

Step 2: Demonstrate that P 1( )is correct.

Step 3: Assume P k( )is correct.

Step 4: Using Step 3 as a guide, show that P k+1( )is correct.

Step 5: As a result, whenever P k( )is true, P 1( )is true and P k+1( )is true.

Class XI Maths 2

As a result, P ( n)is true for all natural integers n, according to the Principle of

Mathematical Induction. Example: Solution: Step 1: LetProve that 2 P n : 2 n > nfor all positive integers n n > n

( )

Step 2: When n =1, 2 >1 1. Hence P 1( )is true.

Step 3: Assume that P k( )is true for any positive integer k, i., 2 k > k ... ( 1 )

Step 4: We shall now prove that P k +1( )is true whenever P k( )is true.

Multiplying both sides of (1) by 2, we get  

or, 2 > k + 1 since k >

or, 2 > k + k

i., 2 + 1 > 2k

2 2 > 2 k

k + 1

( )

Therefore, P k+1( ) is true when P k( ) is true. Hence, by principle of

mathematical induction, P ( n)is true for every positive integer n.

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