89. Mathematical Induction Problems Part 1

All problems are to be solved using mathematical induction.

1. Show that $$1^2 + 2^2 + \ldots + n^2 = \frac{n(n + 1)(2n + 1)}{6}$$

2. Show that $$\frac{1}{1.2} + \frac{1}{2.3} + \ldots + \frac{1}{n(n + 1)} = \frac{n}{n + 1}$$

3. Show that $$1^3 + 2^3 + \ldots + n^3 = \left(\frac{n(n + 1)}{2}\right)^2$$

4. Show that $$1.3 + 2.3^2 + \ldots + n.3^n = \frac{(2n - 1)3^{n + 1} + 3}{4}$$

5. Show that $$\cos\alpha + \cos 2\alpha + \ldots + \cos n\alpha = \sin \frac{n\alpha}{2}\text{cosec}\frac{\alpha}{2}\cos\frac{(n + 1)\alpha}{2}$$

6. Show that $$\tan^{-1}\frac{1}{3} + \tan^{-1}\frac{1}{7} + \ldots + \tan^{-1}\frac{1}{n^2 + n + 1} = \tan^{-1}\frac{n}{n + 2}$$

7. Show that $${}^nC_1 + 2.{}^nC_2 + \ldots + n.{}^nC_n = n.2^{n - 1}$$

8. If $$u_1 = 1, u_2 = 1$$ and $$u_{n + 2} = u_{n + 1} + u_n, n\geq 1.$$ $$u_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1 + \sqrt{5}}{2}\right)^n - \left(\frac{1 +- \sqrt{5}}{2}\right)^n\right]~\forall~ n \geq 1$$

9. Show that $$11^{n + 2} + 12^{2n + 1},$$ where $$n$$ is a natural number, is divisible by $$133.$$

10. If $$p$$ be a natural number, show that $$p^{n + 1} + (p + 1)^{2n - 1}$$ is divisible by $$p^2 + p + 1$$ for every positive integer $$n.$$

11. Show that $$2^n > 2n + 1~\forall~n>2$$

12. Show that $$n^4 < 10^n~\forall~n \geq 2$$

13. Show that $$1^3 + 3^3 + \ldots + (2n - 1)^3 = n^2(2n^2 - 1)$$

14. Show that $$3.2^2 + 3^2.2^3 + \ldots + 3^n.2^{n + 1} = \frac{12}{5}(6^n - 1)$$

15. Show that $$\frac{1}{1.4} + \frac{1}{4.7} + \ldots + \frac{1}{(3n - 2)(3n + 1)} = \frac{n}{3n + 1}$$

16. Show that $$(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$$

17. Show that $$\cos\theta.\cos 2\theta \ldots \cos 2^{n - 1}\theta = \frac{\sin 2^n\theta}{2^n\sin\theta}$$

18. Show that $$\sin\alpha + \sin 2\alpha + \ldots + \sin n\alpha = \frac{\sin \frac{n\alpha}{2}}{\sin \frac{\alpha}{2}}\sin \frac{n + 1}{2}\alpha$$

19. If $$a_1 = 1$$ and $$a_{n + 1} = \frac{a_n}{n + 1}, n\geq 1$$ show that $$a_{n + 1} = \frac{1}{(n + 1)!}$$

20. If $$a_1 = 1, a_2 = 5$$ and $$a_{n + 2} = 5a_{n + 1} - 6a_n, n\geq 1$$ show that $$a_n = 3^n - 2^n$$

21. If $$u_0 = 2, u_1 = 3$$ and $$u_{n + 1} = 3u_n - 2u_{n - 1},$$ show that $$u_n = 2^n + 1, n\in N$$

22. If $$a_0 = 0, a_1 = 1$$ and $$a_{n + 1} = 3a_n - 2a_{n - 1},$$ show that $$a_n = 2^n - 1$$

23. If $$A_1 = \cos\theta, A_2=\cos 2\theta$$ and for every natural number $$m > 2, A_m = 2A_{m - 1}\cos\theta -A_{m - 2},$$ prove that $$A_n = \cos n\theta$$

24. For any positive number $$n,$$ show by induction that $$(2\cos\theta - 1)(2\cos 2\theta - 1)\ldots(2\cos 2^{n - 1}\theta - 1) = \frac{2\cos 2^n\theta + 1}{2\cos\theta + 1}$$

25. Show hat $$\tan^{-1}\frac{x}{1.2 + x^2} + \tan^{-1}\frac{x}{2.3 + x^2} + \ldots + \tan^{-1}\frac{x}{n(n + 1) + x^2} = \tan^{-1}x - \tan^{-1}\frac{x}{n + 1}, x\in R$$

26. Prove that $$3 + 33 + \ldots + \frac{33\ldots3}{n~\text{digits}} = \frac{10^{n + 1} - 9n -10}{27}$$

27. Show that $$\int_{0}^{\pi}\frac{\sin(2n + 1)x}{\sin x}dx = \pi$$

28. Show that $$\int_{0}^{\pi}\frac{\sin^2 nx}{\sin^2x}dx = n\pi$$

29. Show that $$\int_{0}^{\frac{\pi}{2}}\frac{\sin^2 nx}{\sin^2x}dx = 1 + \frac{1}{3} + \ldots + \frac{1}{2n - 1}$$

30. For $$n\in N, n(n + 1)(n + 5)$$ is divisible by $$6$$

31. For $$n\in N, n^3 + (n + 1)^3 + (n + 2)^3$$ is divisible by $$9$$

32. Show that $$n(n^2 + 20)$$ is divisible by $$48,$$ where $$n$$ is a positive even integer.

33. For $$n\in N, 4^n - 3n - 1$$ is divisible by $$9$$

34. For $$n\in N, 3^{2n} - 1$$ is divisible by $$8$$

35. For $$n\in N, 5.2^{3n - 2} + 3^{3n - 1}$$ is divisible by $$19$$

36. For $$n\in N, 7^{2n} + 2^{3n - 3}.3^{n - 1}$$ is divisible by $$25$$

37. For $$n\in N, 10^n+ 3.4^{n + 2} + 5$$ is divisible by $$9$$

38. For $$n\in N, 3^{4n + 2} + 5^{2n + 1}$$ is divisible by $$14$$

39. For $$n\in N, 3^{2n + 2} - 8n - 9$$ is divisible by $$64$$

40. For $$n\in N, n^7 - n$$ is divisible by $$7$$

41. $$\frac{n^3}{3} + n^2 + \frac{5}{3}n + 1$$ is a natural number.

42. $$x^n + y^n$$ is divisible by $$x + y,$$ where $$n$$ is any odd positive integer.

43. Prove that $$x(x^{n - 1} - na^{n - 1}) + a^n(n - 1)$$ is divisible by $$(x - a)^2$$ for all positive integers $$n > 1$$

44. $$\frac{n^5}{5} + \frac{n^3}{3} + \frac{7n}{15}$$ is a natural number.

45. $$\frac{n^7}{7} + \frac{n^5}{5} + \frac{2n^3}{3} - \frac{n}{105}$$ is an integer.

46. Show that $$2^n > 2$$

47. Show that $$2^n > n^2, n\geq 5$$

48. Show that $$1 + 2 + \ldots + n < \frac{1}{8}(2n + 1)^2$$

49. $$n^n < (n!)^2, n > 2$$

50. $$n! > 2^n, n > 3$$

51. $$\frac{1}{n + 1} + \frac{1}{n + 2} + \ldots + \frac{1}{2n} > \frac{13}{24}, n > 1$$