NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction — Exercise 4.20

Question 1:1 + 3 + 32 + ... + 3n – 1 = (3 1) 2 n − .

Question 2:13 + 23 + 33 + … +n3 = 2 ( 1) 2 æ n n + ö ç ÷ è ø .

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Question 3:1 1 1 2 1 (1 2) (1 2 3) (1 2 3 ) ( 1) n ... ...n n + + + + = + + + + + + + .

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Question 4:1.2.3 + 2.3.4 +…+ n(n+1) (n+2) = ( 1) ( 2) ( 3) 4 n n + n + n + .

Question 5:1.3 + 2.32 + 3.33 +…+ n.3n = 1 (2 1)3 3 4 n n − + + .

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Question 6:1.2 + 2.3 + 3.4 +…+ n.(n+1) = ( 1) ( 2) 3 é n n + n + ù êë úû .

Question 7:1.3 + 3.5 + 5.7 +…+ (2n–1) (2n+1) = 2 (4 6 1) 3 n n + n − .

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Question 8:1.2 + 2.22 + 3.23 + ...+n.2n = (n–1) 2n + 1 + 2.

Question 9:1 1 1 1 1 ... 1 2 4 8 2 2 n n + + + + = − .

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Question 10:1 1 1 1 ... 2.5 5.8 8.11 (3 1) (3 2) (6 4) n n n n + + + + = − + + .

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Question 11:1 1 1 1 ( 3) ... 1.2.3 2.3.4 3.4.5 ( 1) ( 2) 4( 1) ( 2) n n n n n n n + + + + + = + + + + .

Question 12:a + ar + ar2 +…+ arn-1 = ( 1) 1 n a r r − − .

Question 13:2 2 3 5 7 (2 1) 1 1 1 ... 1 ( 1) 1 4 9 n n n æ ö æ ö æ ö æ + ö ç + ÷ ç + ÷ ç + ÷ ç + ÷ = + è ø è ø è ø è ø .

Question 14:1 1 1 1 1 1 1 ... 1 ( 1) 1 2 3 n n æ + ö æ + ö æ + ö æ + ö = + ç ÷ ç ÷ ç ÷ ç ÷ è ø è ø è ø è ø .

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Question 15:12 + 32 + 52 + …+ (2n–1)2 = (2 1) (2 1) 3 n n − n + .

EXERCISE 4.1Prove the following by using the princ

Question 16:1 1 1 1 ... 1.4 4.7 7.10 (3 2)(3 1) (3 1) n n n n + + + + = − + + .

Question 17:1 1 1 1 ... 3.5 5.7 7.9 (2 1)(2 3) 3(2 3) n n n n + + + + = + + + .

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Question 18:1 + 2 + 3 +…+ n < 1 8 (2n + 1)2.

Question 19:n (n + 1) (n + 5) is a multiple of 3.

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Question 20:102n – 1 + 1 is divisible by 11.

Question 21:x2n – y2n is divisible by x + y.

Principle of Mathematical Induction ncert solutions class 11

Question 22:32n+2 – 8n – 9 is divisible by 8.

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Question 23:41n – 14n is a multiple of 27.

Question 24:(2n + 7) < (n + 3)2.