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

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 solutions for Class 11 Mathematics chapter 12 - Mathematical Induction

RD Sharma solutions for Class 11 Maths 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 Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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

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