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

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Chapters

RD Sharma Mathematics Class 11

Mathematics Class 11

Chapter 12: Mathematical Induction

Ex. 12.10Ex. 12.20

Chapter 12: Mathematical Induction Exercise 12.10 solutions [Page 3]

Ex. 12.10 | Q 1 | Page 3

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

Ex. 12.10 | 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.

Ex. 12.10 | 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.

 
Ex. 12.10 | 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.

 
Ex. 12.10 | Q 5 | Page 3

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

 
Ex. 12.10 | 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.

Ex. 12.10 | 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.

Chapter 12: Mathematical Induction Exercise 12.20 solutions [Pages 27 - 29]

Ex. 12.20 | Q 1 | Page 27

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

Ex. 12.20 | Q 2 | Page 27

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

 
Ex. 12.20 | Q 3 | Page 27

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

 
Ex. 12.20 | Q 4 | Page 27

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

Ex. 12.20 | Q 5 | Page 27

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

 
Ex. 12.20 | Q 6 | Page 27

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

 

Ex. 12.20 | Q 7 | Page 27

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

Ex. 12.20 | Q 8 | Page 27

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

Ex. 12.20 | Q 9 | Page 27

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

Ex. 12.20 | Q 10 | Page 27

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

 
Ex. 12.20 | Q 11 | Page 27

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

 
Ex. 12.20 | Q 12 | Page 27

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

 
Ex. 12.20 | Q 13 | Page 27

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

 
Ex. 12.20 | Q 14 | Page 27

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

 
Ex. 12.20 | Q 15 | Page 27

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

Ex. 12.20 | Q 16 | Page 27

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

 
Ex. 12.20 | Q 17 | Page 27

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

 
Ex. 12.20 | Q 18 | Page 28

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

 

Ex. 12.20 | Q 19 | Page 28

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

Ex. 12.20 | Q 20 | Page 28

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

 
Ex. 12.20 | Q 21 | Page 28

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

 
Ex. 12.20 | Q 22 | Page 28

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

 
Ex. 12.20 | Q 23 | Page 28

(ab)n = anbn for all n ∈ N. 

 
Ex. 12.20 | Q 24 | Page 28

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

 
Ex. 12.20 | Q 25 | Page 28

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

 
Ex. 12.20 | Q 26 | Page 28

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

 
Ex. 12.20 | Q 27 | Page 28

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

 
Ex. 12.20 | 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 \[\frac{a_n - \sqrt{A}}{a_n + \sqrt{A}} = \left( \frac{a_1 - \sqrt{A}}{a_1 + \sqrt{A}} \right) 2^{n - 1}\]

 
Ex. 12.20 | Q 29 | Page 28

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

  
Ex. 12.20 | Q 30 | Page 28

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

 
Ex. 12.20 | Q 31 | Page 28

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

 
Ex. 12.20 | 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.  

 

Ex. 12.20 | 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

 

Ex. 12.20 | 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  \[0 < x < \frac{\pi}{2}\]

 

Ex. 12.20 | 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

Ex. 12.20 | Q 36 | Page 28

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

Ex. 12.20 | 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 ∈ 

 

Ex. 12.20 | Q 38 | Page 28

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

 
Ex. 12.20 | Q 39 | Page 28
\[\sin x + \sin 3x + . . . + \sin (2n - 1)x = \frac{\sin^2 nx}{\sin x}\]

 

Ex. 12.20 | Q 40 | Page 29
\[\text{ Prove that } \cos\alpha + \cos\left( \alpha + \beta \right) + \cos\left( \alpha + 2\beta \right) + . . . + \cos\left[ \alpha + \left( n - 1 \right)\beta \right] = \frac{\cos\left\{ \alpha + \left( \frac{n - 1}{2} \right)\beta \right\}\sin\left( \frac{n\beta}{2} \right)}{\sin\left( \frac{\beta}{2} \right)} \text{ for all n } \in N .\]

 

Ex. 12.20 | Q 41 | Page 29
\[\text{ Prove that }  \frac{1}{n + 1} + \frac{1}{n + 2} + . . . + \frac{1}{2n} > \frac{13}{24}, \text{ for all natural numbers } n > 1 .\]

 

Ex. 12.20 | Q 42 | Page 29

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

Ex. 12.20 | Q 43 | Page 29

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

Ex. 12.20 | 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 \[S_n = \binom{\frac{n \left( n + 1 \right)^2}{2}, \text{ if n is even} }{\frac{n^2 \left( n + 1 \right)}{2}, \text{ if n is odd } }\]

 

Ex. 12.20 | Q 45 | Page 29

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

 
Ex. 12.20 | Q 46 | Page 29

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

Ex. 12.20 | Q 47 | Page 29

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

Ex. 12.20 | Q 48 | Page 29

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

Ex. 12.20 | Q 49 | Page 29
\[\text{ Using principle of mathematical induction, prove that } \sqrt{n} < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + . . . + \frac{1}{\sqrt{n}} \text{ for all natural numbers } n \geq 2 .\]

 

Ex. 12.20 | 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

Ex. 12.10Ex. 12.20

RD Sharma Mathematics Class 11

Mathematics Class 11

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

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

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