Principle of Mathematical Induction Class 11 NCERT Solutions,

In this pdf file you can see answers of following Questions

EXERCISE 4.1

Prove the following by using the principle of mathematical induction for all n ∈ N:

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

Question 2. 13 + 23 + 33 + … +n3 = 2 ( 1) 2 n n + .

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

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 = (2 1)3 1 3 4 n − n+ + .

Question 6. 1.2 + 2.3 + 3.4 +…+ n.(n+1) = ( 1)( 2) 3 n n + n +.

Question 7. 1.3 + 3.5 + 57 +…+ (2n–1) (2n+1) = (4 2 6 1) 3 n n + n − .

Question 8. 1.2 + 2.22 + 3.22 + ...+n.2n = (n–1) 2n + 1 + 2.

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

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

Question 11. 1 1 1 ... 1 ( 3) 1.2.3 2.3.4 3.4.5 ( 1)( 2) 4( 1)( 2)

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

Question 13. 2 2 1 3 1 5 1 7 ... 1 (2 1) ( 1) 1 4 9 n n

Question 14. 12 + 32 + 52 + …+ (2n–1)2 = (2 1)(2 1) 3 n n − n + .

Question 15. 1 1 1 ... 1 1.4 4.7 7.10 (3 2)(3 1) (3 1) n n n n + + + + = − + + .

Question 16. 1 1 1 ... 1 3.5 5.7 7.9 (2 1)(2 3) 3(2 3) n n n n + + + + = + + + .

Question 17. 1 + 2 + 3 +…+ n < 1 8 (2n + 1)2.

Question 18. n (n + 1) (n + 5) is a multiple of 3.

Question 19. 102n – 1 + 1 is divisible by 11.

Question 20. x2n – y2n is divisible by x + y.

Question 21. 32n+2 – 8n – 9 is divisible by 8.

Question 22. 41n – 14n is a multiple of 27.

Question 23."> (2n + 7) < (n + 3)2.

NCERT Books Free Pdf Download for Class 5, 6, 7, 8, 9, 10 , 11, 12 Hindi and English Medium

 Mathematics Biology Psychology Chemistry English Economics Sociology Hindi Business Studies Geography Science Political Science Statistics Physics Accountancy