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RD Sharma solutions for Class 11 Mathematics Textbook chapter 12 - Mathematical Induction [Latest edition]

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Class 11 Mathematics Textbook - Shaalaa.com
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Chapter 12: Mathematical Induction

Exercise 12.1Exercise 12.2
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Exercise 12.1 [Page 3]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 12 Mathematical InductionExercise 12.1 [Page 3]

Exercise 12.1 | Q 1 | Page 3

If P (n) is the statement "n(n + 1) is even", then what is P(3)?

Exercise 12.1 | Q 2 | Page 3

If P (n) is the statement "n3 + n is divisible by 3", prove that P (3) is true but P (4) is not true.

Exercise 12.1 | Q 3 | Page 3

If P (n) is the statement "2n ≥ 3n" and if P (r) is true, prove that P (r + 1) is true.

 
Exercise 12.1 | Q 4 | Page 3

If P (n) is the statement "n2 + n is even", and if P (r) is true, then P (r + 1) is true.

 
Exercise 12.1 | Q 5 | Page 3

Given an example of a statement P (n) such that it is true for all n ∈ N.

 
Exercise 12.1 | Q 6 | Page 3

If P (n) is the statement "n2 − n + 41 is prime", prove that P (1), P (2) and P (3) are true. Prove also that P (41) is not true.

Exercise 12.1 | Q 7 | Page 3

Give an example of a statement P(n) which is true for all n ≥ 4 but P(1), P(2) and P(3) are not true. Justify your answer.

Exercise 12.2 [Pages 27 - 29]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 12 Mathematical InductionExercise 12.2 [Pages 27 - 29]

Exercise 12.2 | Q 1 | Page 27

1 + 2 + 3 + ... + n =  \[\frac{n(n + 1)}{2}\] i.e. the sum of the first n natural numbers is \[\frac{n(n + 1)}{2}\] .

Exercise 12.2 | Q 2 | Page 27

12 + 22 + 32 + ... + n2 =\[\frac{n(n + 1)(2n + 1)}{6}\] .

 
Exercise 12.2 | Q 3 | Page 27

1 + 3 + 32 + ... + 3n−1 = \[\frac{3^n - 1}{2}\]

 
Exercise 12.2 | Q 4 | Page 27

\[\frac{1}{1 . 2} + \frac{1}{2 . 3} + \frac{1}{3 . 4} + . . . + \frac{1}{n(n + 1)} = \frac{n}{n + 1}\]

Exercise 12.2 | Q 5 | Page 27

1 + 3 + 5 + ... + (2n − 1) = n2 i.e., the sum of first n odd natural numbers is n2.

 
Exercise 12.2 | Q 6 | Page 27

\[\frac{1}{2 . 5} + \frac{1}{5 . 8} + \frac{1}{8 . 11} + . . . + \frac{1}{(3n - 1)(3n + 2)} = \frac{n}{6n + 4}\]

 

Exercise 12.2 | Q 7 | Page 27

\[\frac{1}{1 . 4} + \frac{1}{4 . 7} + \frac{1}{7 . 10} + . . . + \frac{1}{(3n - 2)(3n + 1)} = \frac{n}{3n + 1}\]

Exercise 12.2 | Q 8 | Page 27

\[\frac{1}{3 . 5} + \frac{1}{5 . 7} + \frac{1}{7 . 9} + . . . + \frac{1}{(2n + 1)(2n + 3)} = \frac{n}{3(2n + 3)}\]

Exercise 12.2 | Q 9 | Page 27

\[\frac{1}{3 . 7} + \frac{1}{7 . 11} + \frac{1}{11 . 5} + . . . + \frac{1}{(4n - 1)(4n + 3)} = \frac{n}{3(4n + 3)}\] 

Exercise 12.2 | Q 10 | Page 27

1.2 + 2.22 + 3.23 + ... + n.2= (n − 1) 2n+1+2

 
Exercise 12.2 | Q 11 | Page 27

2 + 5 + 8 + 11 + ... + (3n − 1) = \[\frac{1}{2}n(3n + 1)\]

 
Exercise 12.2 | Q 12 | Page 27

1.3 + 2.4 + 3.5 + ... + n. (n + 2) = \[\frac{1}{6}n(n + 1)(2n + 7)\]

 
Exercise 12.2 | Q 13 | Page 27

1.3 + 3.5 + 5.7 + ... + (2n − 1) (2n + 1) =\[\frac{n(4 n^2 + 6n - 1)}{3}\]

 
Exercise 12.2 | Q 14 | Page 27

1.2 + 2.3 + 3.4 + ... + n (n + 1) = \[\frac{n(n + 1)(n + 2)}{3}\]

 
Exercise 12.2 | Q 15 | Page 27

\[\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + . . . + \frac{1}{2^n} = 1 - \frac{1}{2^n}\]

Exercise 12.2 | Q 16 | Page 27

12 + 32 + 52 + ... + (2n − 1)2 = \[\frac{1}{3}n(4 n^2 - 1)\]

 
Exercise 12.2 | Q 17 | Page 27

a + ar + ar2 + ... + arn−1 =  \[a\left( \frac{r^n - 1}{r - 1} \right), r \neq 1\]

 
Exercise 12.2 | Q 18 | Page 28

a + (a + d) + (a + 2d) + ... (a + (n − 1) d) = \[\frac{n}{2}\left[ 2a + (n - 1)d \right]\]

 

Exercise 12.2 | Q 19 | Page 28

52n −1 is divisible by 24 for all n ∈ N.

Exercise 12.2 | Q 20 | Page 28

32n+7 is divisible by 8 for all n ∈ N.

 
Exercise 12.2 | Q 21 | Page 28

52n+2 −24n −25 is divisible by 576 for all n ∈ N.

 
Exercise 12.2 | Q 22 | Page 28

32n+2 −8n − 9 is divisible by 8 for all n ∈ N.

 
Exercise 12.2 | Q 23 | Page 28

(ab)n = anbn for all n ∈ N. 

 
Exercise 12.2 | Q 24 | Page 28

n(n + 1) (n + 5) is a multiple of 3 for all n ∈ N.

 
Exercise 12.2 | Q 25 | Page 28

72n + 23n−3. 3n−1 is divisible by 25 for all n ∈ N.

 
Exercise 12.2 | Q 26 | Page 28

2.7n + 3.5n − 5 is divisible by 24 for all n ∈ N.

Exercise 12.2 | Q 27 | Page 28

11n+2 + 122n+1 is divisible by 133 for all n ∈ N.

 
Exercise 12.2 | Q 28 | Page 28

Given \[a_1 = \frac{1}{2}\left( a_0 + \frac{A}{a_0} \right), a_2 = \frac{1}{2}\left( a_1 + \frac{A}{a_1} \right) \text{ and }  a_{n + 1} = \frac{1}{2}\left( a_n + \frac{A}{a_n} \right)\] for n ≥ 2, where a > 0, A > 0.
Prove that \[\frac{a_n - \sqrt{A}}{a_n + \sqrt{A}} = \left( \frac{a_1 - \sqrt{A}}{a_1 + \sqrt{A}} \right) 2^{n - 1}\]

 
Exercise 12.2 | Q 29 | Page 28

Prove that n3 - 7+ 3 is divisible by 3 for all n \[\in\] N .

  
Exercise 12.2 | Q 30 | Page 28

Prove that 1 + 2 + 22 + ... + 2n = 2n+1 - 1 for all \[\in\] N .

 
Exercise 12.2 | Q 31 | Page 28

7 + 77 + 777 + ... + 777 \[{. . . . . . . . . . .}_{n - \text{ digits } } 7 = \frac{7}{81}( {10}^{n + 1} - 9n - 10)\]

 
Exercise 12.2 | Q 32 | Page 28
\[\frac{n^7}{7} + \frac{n^5}{5} + \frac{n^3}{3} + \frac{n^2}{2} - \frac{37}{210}n\] is a positive integer for all n ∈ N.  

 

Exercise 12.2 | Q 33 | Page 28
\[\frac{n^{11}}{11} + \frac{n^5}{5} + \frac{n^3}{3} + \frac{62}{165}n\] is a positive integer for all n ∈ N

 

Exercise 12.2 | Q 34 | Page 28
\[\frac{1}{2}\tan\left( \frac{x}{2} \right) + \frac{1}{4}\tan\left( \frac{x}{4} \right) + . . . + \frac{1}{2^n}\tan\left( \frac{x}{2^n} \right) = \frac{1}{2^n}\cot\left( \frac{x}{2^n} \right) - \cot x\] for all n ∈ and  \[0 < x < \frac{\pi}{2}\]

 

Exercise 12.2 | Q 35 | Page 28

Let P(n) be the statement : 2n ≥ 3n. If P(r) is true, show that P(r + 1) is true. Do you conclude that P(n) is true for all n ∈ N

Exercise 12.2 | Q 36 | Page 28

\[\frac{(2n)!}{2^{2n} (n! )^2} \leq \frac{1}{\sqrt{3n + 1}}\]  for all n ∈ N .

Exercise 12.2 | Q 37 | Page 28
\[1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + . . . + \frac{1}{n^2} < 2 - \frac{1}{n}\] for all n ≥ 2, n ∈ 

 

Exercise 12.2 | Q 38 | Page 28

x2n−1 + y2n−1 is divisible by x + y for all n ∈ N.

 
Exercise 12.2 | Q 39 | Page 28
\[\sin x + \sin 3x + . . . + \sin (2n - 1)x = \frac{\sin^2 nx}{\sin x}\]

 

Exercise 12.2 | Q 40 | Page 29
\[\text{ Prove that } \cos\alpha + \cos\left( \alpha + \beta \right) + \cos\left( \alpha + 2\beta \right) + . . . + \cos\left[ \alpha + \left( n - 1 \right)\beta \right] = \frac{\cos\left\{ \alpha + \left( \frac{n - 1}{2} \right)\beta \right\}\sin\left( \frac{n\beta}{2} \right)}{\sin\left( \frac{\beta}{2} \right)} \text{ for all n } \in N .\]

 

Exercise 12.2 | Q 41 | Page 29
\[\text{ Prove that }  \frac{1}{n + 1} + \frac{1}{n + 2} + . . . + \frac{1}{2n} > \frac{13}{24}, \text{ for all natural numbers } n > 1 .\]

 

Exercise 12.2 | Q 42 | Page 29

\[\text{ Given }  a_1 = \frac{1}{2}\left( a_0 + \frac{A}{a_0} \right), a_2 = \frac{1}{2}\left( a_1 + \frac{A}{a_1} \right) \text{ and } a_{n + 1} = \frac{1}{2}\left( a_n + \frac{A}{a_n} \right) \text{ for }  n \geq 2, \text{ where } a > 0, A > 0 . \]
\[\text{ Prove that } \frac{a_n - \sqrt{A}}{a_n + \sqrt{A}} = \left( \frac{a_1 - \sqrt{A}}{a_1 + \sqrt{A}} \right) 2^{n - 1} .\]

Exercise 12.2 | Q 43 | Page 29

\[\text{ Let } P\left( n \right) \text{ be the statement } : 2^n \geq 3n . \text{ If } P\left( r \right) \text{ is true, then show that } P\left( r + 1 \right) \text{ is true . Do you conclude that } P\left( n \right)\text{  is true for all n }  \in N?\]

Exercise 12.2 | Q 44 | Page 29

Show by the Principle of Mathematical induction that the sum Sn of then terms of the series  \[1^2 + 2 \times 2^2 + 3^2 + 2 \times 4^2 + 5^2 + 2 \times 6^2 + 7^2 + . . .\] is given by \[S_n = \binom{\frac{n \left( n + 1 \right)^2}{2}, \text{ if n is even} }{\frac{n^2 \left( n + 1 \right)}{2}, \text{ if n is odd } }\]

 

Exercise 12.2 | Q 45 | Page 29

Prove that the number of subsets of a set containing n distinct elements is 2n, for all n \[\in\] N .

 
Exercise 12.2 | Q 46 | Page 29

\[\text{ A sequence }  a_1 , a_2 , a_3 , . . . \text{ is defined by letting }  a_1 = 3 \text{ and } a_k = 7 a_{k - 1} \text{ for all natural numbers } k \geq 2 . \text{ Show that } a_n = 3 \cdot 7^{n - 1} \text{ for all } n \in N .\]

Exercise 12.2 | Q 47 | Page 29

\[\text { A sequence  } x_1 , x_2 , x_3 , . . . \text{ is defined by letting } x_1 = 2 \text{ and }  x_k = \frac{x_{k - 1}}{k} \text{ for all natural numbers } k, k \geq 2 . \text{ Show that }  x_n = \frac{2}{n!} \text{ for all } n \in N .\]

Exercise 12.2 | Q 48 | Page 29

\[\text{ A sequence } x_0 , x_1 , x_2 , x_3 , . . . \text{ is defined by letting } x_0 = 5 and x_k = 4 + x_{k - 1}\text{  for all natural number k . } \]
\[\text{ Show that } x_n = 5 + 4n \text{ for all n }  \in N \text{ using mathematical induction .} \]

Exercise 12.2 | Q 49 | Page 29
\[\text{ Using principle of mathematical induction, prove that } \sqrt{n} < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + . . . + \frac{1}{\sqrt{n}} \text{ for all natural numbers } n \geq 2 .\]

 

Exercise 12.2 | Q 50 | Page 29

\[\text{ The distributive law from algebra states that for all real numbers}  c, a_1 \text{ and }  a_2 , \text{ we have }  c\left( a_1 + a_2 \right) = c a_1 + c a_2 . \]
\[\text{ Use this law and mathematical induction to prove that, for all natural numbers, } n \geq 2, if c, a_1 , a_2 , . . . , a_n \text{ are any real numbers, then } \]
\[c\left( a_1 + a_2 + . . . + a_n \right) = c a_1 + c a_2 + . . . + c a_n\]

Chapter 12: Mathematical Induction

Exercise 12.1Exercise 12.2
Class 11 Mathematics Textbook - Shaalaa.com

RD Sharma solutions for Class 11 Mathematics Textbook chapter 12 - Mathematical Induction

RD Sharma solutions for Class 11 Mathematics Textbook chapter 12 (Mathematical Induction) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 11 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics Textbook chapter 12 Mathematical Induction are Motivation, Principle of Mathematical Induction.

Using RD Sharma Class 11 solutions Mathematical Induction exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

Get the free view of chapter 12 Mathematical Induction Class 11 extra questions for Class 11 Mathematics Textbook and can use Shaalaa.com to keep it handy for your exam preparation

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