NCERT Solutions for Class 11 Maths Maths Chapter 4

Principle of Mathematical Induction Class 11

Chapter 4 Principle of Mathematical Induction Exercise 4.1 Solutions

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Exercise 4.1 : Solutions of Questions on Page Number : 94

Q1 :  

 Prove the following by using the principle of mathematical induction for all nN:

 


Answer :

Let the given statement be P(n), i.e.,

P(n): 1 + 3 + 32 + …+ 3n–1 =

For n = 1, we have

P(1): 1 =, which is true.

Let P(k) be true for some positive integer k, i.e.,

We shall now prove that P(k + 1) is true.

Consider

1 + 3 + 32 + … + 3k–1 + 3(k+1) – 1

= (1 + 3 + 32 +… + 3k–1) + 3k


 

 

 

 

 

 

 

 

 

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

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Q2 :  

Prove the following by using the principle of mathematical induction for all nN:


Answer :

Let the given statement be P(n), i.e.,

P(n):

For n = 1, we have

P(1): 13 = 1 =, which is true.

Let P(k) be true for some positive integer k, i.e.,

We shall now prove that P(k + 1) is true.

Consider

13 + 23 + 33 + … + k3 + (k + 1)3

= (13 + 23 + 33 + …. + k3) + (k + 1)3

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

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Q3 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q4 :  

Prove the following by using the principle of mathematical induction for all nN: 1.2.3 + 2.3.4 + … + n(n + 1) (n + 2) =


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Q5 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q6 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q7 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q8 :  

Prove the following by using the principle of mathematical induction for all nN: 1.2 + 2.22 + 3.22 + … + n.2n = (n - 1) 2n+1 + 2


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Q9 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q10 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q11 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q12 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q13 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q14 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q15 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q16 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q17 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q18 :  

Prove the following by using the principle of mathematical induction for all nN:


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Q19 :  

Prove the following by using the principle of mathematical induction for all nN: n (n + 1) (n + 5) is a multiple of 3.


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Q20 :  

Prove the following by using the principle of mathematical induction for all nN: 102n - 1 + 1 is divisible by 11.


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Q21 :  

Prove the following by using the principle of mathematical induction for all nN: x2n - y2n is divisible by x + y.


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Q22 :  

Prove the following by using the principle of mathematical induction for all nN: 32n + 2 - 8n - 9 is divisible by 8.


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Q23 :  

Prove the following by using the principle of mathematical induction for all nN: 41n - 14n is a multiple of 27.


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Q24 :  

Prove the following by using the principle of mathematical induction for all

(2n +7) < (n + 3)2


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<< Previous Chapter 3 : Trigonometric Functions Next Chapter 5 : Complex Numbers and Quadratic Equations >>

Maths - Maths : CBSE NCERT Exercise Solutions for Class 11th for Principle of Mathematical Induction ( Exercise 4.1 ) will be available online in PDF book form soon. The solutions are absolutely Free. Soon you will be able to download the solutions.

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