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. \(\frac{1}{n + 1} + \frac{1}{n + 2} + \ldots + \frac{1}{2n} > \frac{13}{24}, n > 1\)