# 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.

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