92. Binomials, Multinomials and Expansions Problems

  1. Expand \(\left(x + \frac{1}{x}\right)^5\)

  2. Use the binomial theorem to find the exact value of \((10.1)^5\)

  3. Simplify \((x + \sqrt{x - 1})^6 + (x - \sqrt{x - 1})^6\)

  4. If \(A\) be the sum of odd terms and \(B\) be the sum of even terms in the expansion of \((x + a)^n,\) prove that \(A^2 - B^2 = (x^2 - a^2)^n\)

  5. Solve following:

    1. If \(n\) be a positive integer, then prove that the integral part of \((7 + 4\sqrt{3})^n\) is an odd number.

    2. If \((7 + 4\sqrt{3})^n = \alpha + \beta,\) where \(\alpha\) is a positive integer and \(\beta\) a proper fraction, then prove that \((1 - \beta)(\alpha + \beta) = 1\)

  6. Find the coefficient of \(\frac{1}{y^2}\) in \(\left(y + \frac{c^3}{y^2}\right)^{10}\)

  7. Find the coefficient of \(x^9\) in \((1 + 3x + 3x^2 + x^3)^{15}\)

  8. Find the term independent of \(x\) in \(\left(\frac{3}{2}x^2 - \frac{1}{3x}\right)^9\)

  9. Find the term independent of \(x\) in \((1 + x)^m\left(1 + \frac{1}{x}\right)^n\)

  10. Find the coefficient of \(x^{-1}\) in \((1 + 3x^2 + x^4)\left(1 + \frac{1}{x}\right)^n\)

  11. If \(a_r\) denotes the coefficient of \(x^r\) in the expansion \((1 - x)^{2n - 1},\) then prove that \(a_{r - 1} + a_{2n - r} = 0\)

  12. Find the value of \(k\) so that the term independent of \(x\) in \(\left(\sqrt{x} + \frac{k}{x^2}\right)^{10}\) is \(405.\)

  13. Show that there will be no term containing \(x^{2r}\) in the expansion \((x + x^{-2})^{n - 3}\) if \(n - 2r\) is positive but not a multiple of \(3.\)

  14. Show that there will be a term independent of \(x\) in the expansion \((x^a + x^{-b})^n\) only if \(an\) is a multiple of \(a + b.\)

  15. Expand \(\left(x + \frac{1}{x}\right)^7\) by using binomial theorem.

  16. Use binomial theorem to expans \(\left(\frac{2x}{3} - \frac{3}{2x}\right)^6\)

  17. If \((1 + ax)^n = 1 + 8x + 24x^2 + \ldots,\) find \(a\) and \(n.\)

  18. Write \((x + \sqrt{x^2 + 1})6 + (x - \sqrt{x^2 + 1})^6\) as a polynomial of \(x.\)

  19. Find the \(7^{th}\) term in the expansion of \(\left(\frac{4x}{5} - \frac{5}{2x}\right)^9\)

  20. Find the value of \((\sqrt{2} + 1)^6 + (\sqrt{2} - 1)^6.\)

  21. Evaluate \((0.99)^{15}\) correct to four decimal places using binomial theorem.

  22. Evaluate \((999)^3\) using binomial theorem.

  23. Evaluiate \((0.99)^{10}\) correct to four decimal places using binomial theorem.

  24. Find the value of \((1.01)^{10} + (0.99)^{10}\) correct to 7 decimnal places.

  25. If \(A\) be the sum of odd terms and \(B\) be the sum of even terms in the expansion of \((x + a)^n,\) show that \(4AB = (x + a)^{2n} - (x - a)^{2n}.\)

  26. If \(n\) be a positive interger, prove that the integeral part of \((5 + 2\sqrt{6})^n\) is an odd interger.

  27. If \((3 + \sqrt{8})^n = \alpha + \beta,\) where \(\alpha, n\) are positive integers and \(\beta\) is a proper fraction, then prove that \((1 - \beta)(p + \beta) = 1\).

  28. Find the coefficient of \(x\) in the expansion of \(\left(2x - \frac{3}{x}\right)^9\)

  29. Find the coefficient of \(x^7\) in the expansion of \((3x^2 + 5x^{-1})^{11}\)

  30. Find the coefficicent of \(x^9\) in the expansion of \((2x^2 - x^{-1})^{20}\)

  31. Find the coefficient of \(x^{24}\) in the expansion of \((x^2 + 3ax^{-1})^{15}\)

  32. Find the coefficient of \(x^9\) in the expansion of \((x^2 - 3x^-1)^9\)

  33. Find the coefficient of \(x^{-7}\) in the expansion of \(\left(2x - \frac{1}{3x^2}\right)^{11}\)

  34. Find the coefficient of \(x^7\) in the expansion of \(\left(ax^2 + \frac{1}{bx}\right)^{11}\) and the coefficient of \(x^{-7}\) in the expansion of \(\left(ax - \frac{1}{bx^2}\right)^{11}.\) Also find the relation between \(a\) and \(b\) so that the coefficient are equal.

  35. If \(x^p\) occurs in the expansion of \(\left(x^2 + \frac{1}{x}\right)^{2n},\) show that its coefficient is \(\frac{2n!}{\left(\frac{4n - p}{3}\right)!\left(\frac{2n + p}{3}\right)!}\)

  36. Find the term independent of \(x\) in the following binomial expansions:

    1. \(\left(x + \frac{1}{x}\right)^{2n}\)

    2. \(\left(2x^2 + \frac{1}{x}\right)^{15}\)

    3. \(\left(\sqrt{\frac{x}{3}} + \frac{3}{2x^2}\right)^{10}\)

    4. \(\left(2x^2 - \frac{1}{x}\right)^{12}\)

    5. \(\left(2x^2 - \frac{3}{x^3}\right)^{25}\)

    6. \(\left(x^3 - \frac{3}{x^2}\right)^{25}\)

    7. \(\left(x^2 - \frac{3}{x^3}\right)^{10}\)

    8. \(\left(\frac{1}{2}x^{\frac{1}{3}} + x^{-\frac{1}{3}}\right)^8\)

  37. If there is a term independent of \(x\) in \(\left(x + \frac{1}{x^2}\right)^n,\) show that it is equal to \(\frac{n!}{\left(\frac{n}{3}\right)\left(\frac{2n}{3}\right)!}!\)

  38. Prove that in the expansion of \((1 + x)^{m + n},\) coefficients of \(x^m\) and \(x^n\) are equal. \(\forall~m, n > 0, m, n\in N\)

  39. Given that the \(4^{th}\) term in the expansion of \(\left(px + \frac{1}{x}\right)^n\) is \(\frac{5}{2}.\) Find \(n\) and \(p.\)

  40. Find the middle term in the expansion of \(\left(x - \frac{1}{2x}\right)^{12}\)

  41. Find the middle term in the expansion of \(\left(2x^2 - \frac{1}{x}\right)^7\)

  42. FInd the middle term in the expansion of \((1 - 2x + x^2)^n\)

  43. Prove that the middle term in the expansion of \(\left(x + \frac{1}{x}\right)^{2n}\) is \(\frac{1.3.5\ldots (2n - 1)}{n!}.2^n\)

  44. Show that the greatest coefficient in the expansion of \(\left(x + \frac{1}{x}\right)^{2n}\) is \(\frac{1.3.5\ldots (2n - 1)}{n!}.2^n\)

  45. Show that the coefficent of the middle term in \((1 + x)^{2n}\) is equal to the sum of the coefficient of the two middle term in \((1 + x)^{2n - 1}\)

  46. Find the middle term in the expansions of:

    1. \(\left(\frac{2x}{3} - \frac{3y}{2}\right)^{20}\)

    2. \(\left(\frac{2x}{3} - \frac{3}{2x}\right)^{6}\)

    3. \(\left(\frac{x}{y} - \frac{y}{x}\right)^7\)

    4. \((1 + x)^{2n}\)

    5. \((1 - 2x + x^2)^n\)

  47. Find the general and middle term of the expansion \(\left(\frac{x}{y} + \frac{y}{x}\right)^{2n + 1}; n\) being a positive integer show that there is no term free from \(x\) and \(y.\)

  48. Show that the middle term in the expnsions of \(\left(x - \frac{1}{x}\right)^{2n}\) is \(\frac{1.3.5\ldots (2n - 1)}{n!}.(-2)^n\)

  49. If in the expansion of \((1 + x)^{43},\) the coefficient of \((2r + 1)^{th}\) term is equl to the coefficient of \((r + 2)^{th}\) term, find \(r.\)

  50. If the \(r^{th}\) term in the expansion of \((1 + x)^{2n}\) has its coefficientt equal to that of the \((r + 4)^{th}\) term, find \(r\).

  51. If the coefficient of \((2r + 4)^{th}\) term and \((r - 2)^{th}\) term in the expansion of \((1 + x)^{18}\) are equal, find \(r.\)

  52. If the coefficient of \((2r + 5)^{th}\) term and \((r - 6)^{th}\) term in the expnasion of \((1 + x)^{39}\) are equal, find \({}^rC_{12}.\)

  53. Given positive intergers \(r>1, n>2, n\) being even and the coefficient of \(3r^{th}\) term and \((r + 2)^{th}\) term in the expansion of \((1 + x)^{2n}\) are equal, find r.

  54. If the coefficient of \((p + 1)^{th}\) term in the expansion of \((1 + x)^{2n}\) be equal to that of the \((p + 3)^{th}\) term, show that \(p = n - 1\)

  55. Find the two consecutive coefficients in the expansion of \((3x - 2)^{75}\) whose values are equal.

  56. Show that the coefficient of \((r + 1)^{th}\) term in the expansion of \((1 + x)^{n + 1}\) is equal to the sum of the cofficients of the \(r^{th}\) and \((r + 1)^{th}\) term in the expansion of \((1 + x)^n.\)

  57. Find the greatest term in the expansion of \(\left(7 - \frac{10}{3}\right)^{11}\)

  58. Show that if the greatest term in the expansion of \((1 + x)^{2n}\) has also the greatest coefficient \(x\) lies between \(\frac{n}{n + 1}\) and \(\frac{n + 1}{n}.\)

  59. Find the greatest term in the expansion of:

    1. \(\left(2 + \frac{9}{5}\right)^{10}\)

    2. \((4 - 2)^7\)

    3. \((5 + 2)^{13}\)

  60. Find the limits between which \(x\) must lie in order that the greatest term in the expansion of \((1 + x)^{30}\) may have the greatest coefficient.

  61. If \(n\) is a positive integer, then prove that \(6^{2n} - 35n - 1\) is divisible by \(1225\)

  62. Show that \(2^{4n} - 2^n(7n + 1)\) is some multiple of the square of \(14,\) where \(n\) is a positive integer.

  63. Show that \(3^{4n + 1} + 16n - 3\) is divisible by \(256\) if \(n\) is a positive integer.

  64. If \(n\) is a positive integer show that

    1. \(4^n - 3n - 1\) is divisible by \(9\)

    2. \(2^{5n} - 31n - 1\) is divisible by \(961\)

    3. \(3^{2n + 2} -8n -9\) is divisible by \(64\)

    4. \(2^{5n + 5} - 31n - 32\) is divisible by \(961\) if \(n > 1\)

    5. \(3^{2n} - 1 + 24n - 32n^2\) is divisible by \(512\) if \(n > 2\)

  65. If three consecutive coefficient in the expansion of \((1 + x)^n\) be \(165, 330\) and \(462,\) find \(n\) and position of the first coefficient.

  66. If \(a_1, a_2, a_3\) and \(a_4\) be any four consecutive coefficients in the expansion of \((1 + x)^n,\) prove that \(\frac{a_1}{a_1 + a_2} + \frac{a_3}{a_3 + a_4} = \frac{2a_2}{a_2 + a_3}\)

  67. If \(2^{nd}, 3^{rd}\) and \(4^{th}\) terms in the expansion of \((x + y)^n\) be \(240, 720\) and \(1080\) respectively, find \(x, y\) and \(n.\)

  68. If \(a, b, c\) be the three consecutive terms in the expansion of some power of \(1 + x,\) prove that the exponent is \(\frac{2ac + ab + bc}{b^2 - ac}\)

  69. If \(14^{th}, 15^{th}\) and \(16^{th}\) term in the expansion of \((1 + x)^n\) are in A.P., find \(n.\)

  70. If three consecutive terms in the expansion of \((1 + x)^n\) be \(56, 70\) and \(56,\) find \(n\) and the position of the coefficients.

  71. If three consecutive terms in the expansion of \((1 + x)^n\) be \(220, 495\) and \(792,\) find \(n.\)

  72. If \(3^{rd}, 4^{th}\) and \(5^{th}\) terms in the expansion of \((a + x)^n\) be \(84, 280\) and \(560,\) find \(a, x\) and \(n.\)

  73. If \(6^{th}, 7^{th}\) and \(8^{th}\) terms in the expansion of \((x + y)^n\) be \(112, 7\) and \(\frac{1}{4},\) find \(x, y\) and \(n.\)

  74. If \(a, b, c\) and \(d\) be the \(6^{th}, 7^{th}, 8^{th}\) and \(9^{th}\) terms respectively in any binomial expansion, prove that \(\frac{b^2 - ac}{c^2 - bd} = \frac{4a}{3c}\)

  75. If the four consecutive coefficients in any binomial expansion be \(a, b, c, d\) then prove that

    1. \(\frac{a + b}{a}, \frac{b + c}{b}, \frac{c + d}{c}\) are in H.P.

    2. \((bc + ad)(b - c) = 2(ac^2 - b^2d)\)

  76. The coefficient of the \(5^{th}, 6^{th}\) and \(7^{th}\) terms in the expansion of \((1 + x)^n\) are in A.P. Find the value of \(n.\)

  77. If the coefficients of \(2^{nd}, 3^{rd}\) and \(4^{th}\) terms in the expansion of \((1 + x)^{2n}\) are in A.P., show that \(2n^2 - 9n + 7 = 0\)

  78. If the coefficients of \(r^{th}, (r + 1)^{th}\) and \((r + 2)^{th}\) terms in the expansion of \((1 + x)^n\) are in A.P., show that \(n^2 - n(4r + 1) + 4r^2 - 2 = 0\)

  79. If the coefficients of three consecutive terms in the expansion of \((1 + x)^n\) are in the ration \(182:84:30,\) prove that \(n= 18.\)

If \((1 + x)^n = C_0 + C_1x + C_2x^2 + \ldots + C_nx^n,\) prove that

  1. \(C_1 + 2.C_2 + 3.C_3 + \ldots + n.C_n = n.2^{n - 1}\)

  2. \(C_0 + 2.C_1 + 3.C_2 + \ldots + (n + 1).C_n = (n + 2)2^{n - 1}\)

  3. \(C_0 + 3.C_1 + 5.C_2 + \ldots + (2n + 1).C_n = (n + 1)2^n\)

  4. \(C_1 - 2.C_2 + 3.C_3 - 4.C_4 + \ldots + (-1)^nn.C_n = 0\)

  5. \(C_0 + \frac{C_1}{2} + \frac{C_2}{3} + \ldots + \frac{C_n}{n + 1} = \frac{2^{n + 1} - 1}{n + 1}\)

  6. \(C_0 - \frac{C_1}{2} + \frac{C_2}{3} - \ldots + (-1)^n\frac{C_n}{n + 1} = \frac{1}{n + 1}\)

  7. \(\frac{C_1}{2} + \frac{C_3}{4} + \frac{C_5}{6} + \ldots = \frac{2^n - 1}{n + 1}\)

  8. \(2.C_0 + 2^2\frac{C_1}{2} + 2^3\frac{C_2}{3} + \ldots + 2^{n + 1}\frac{C_n}{n + 1} = \frac{3^{n + 1} - 1}{n + 1}\)

  9. \(C_0.C_r + C_1.C_{r + 1} + \ldots + C_{n - r}.C_n = \frac{(2n)!}{(n + r)!(n - r)!}\)

  10. \(C_0^2 + C_1^2 + C_2^2 + \ldots + C_n^2 = \frac{(2n)!}{n!n!}\)

  11. \(\frac{C_1}{C_0} + 2\frac{C_2}{C_1} + 3\frac{C_3}{C_2} + \ldots + n.\frac{C_n}{C_{n - 1}} = \frac{n(n + 1)}{2}\)

  12. \((1 + {}^nC_1 + {}^nC_2 + \ldots + {}^nC_n)^2 = 1 + {}^{2n}C_1 + {}^{2n}C_2 + \ldots + {}^{2n}C_{2n}\)

  13. \((1 + {}^nC_1 + {}^nC_2 + \ldots + {}^nC_n)^5 = 1 + {}^{5n}C_1 + {}^{5n}C_2 + \ldots + {}^{5n}C_{5n}\)

  14. \(C_0 + 5.C_1 + 9.C_2 + \ldots + (4n + 1).C_n = (2n + 1)2^n\)

  15. \(1 - (1 + x)C_1 + (1 + 2x)C_2 - (1 + 3x)C_3 + \ldots = 0\)

  16. \(3.C_1 + 7.C_2 + 11.C_3 + \ldots + (4n - 1)C_n = (2n - 1)2^{n + 1}\)

  17. \(C_0 + \frac{C_2}{3} + \frac{C_4}{5} + \ldots = \frac{2^n}{n + 1}\)

  18. \({}^nC_0{}^{n + 1}C_1 + {}^nC_1{}^{n + 1}C_2 + \ldots + {}^nC_n{}^{n + 1}C_{n + 1} = \frac{(2n + 1)!}{(n + 1)!n!}\)

  19. Prove that the sum of coefficients in the expansion of \((1 + x - 3x^2)^{2163}\) is \(-1\)

  20. If \((1 + x - 2x^2)^6 = 1 + a_1x + a_2x^2 + \ldots + a_{12}x^{12},\) show that \(a_2 + a_4 + a_6 + \ldots + a_{12} = 31\)

  21. Find the sum of rational terms in the expansion of \((2 + \sqrt[5]{3})^{10}\)

  22. Find the fractional part of \(\frac{2^{4n}}{15}.\)

  23. Show that the integer just above \((\sqrt{3} + 1)^{2n}\) is divisible by \(2^{n + 1}, \forall n \in N.\)

  24. Let \(R = (5\sqrt{5} + 11)^{2n + 1}\) and \(f = R - [R]\) where \([]\) denotes the greatest integre function. Prove that \(Rf = 4^{2n + 1}\)

  25. Show that \((101)^{50} > (100)^{50} + (99)^{50}\)

  26. Find the sum of the series \(\sum_{r = 0}^n (-1)^r.{}^nC_r\left[\frac{1}{2^3} + \frac{3}{2^{2r}} + \frac{7^r}{2^{3r}} + \ldots~\text{to}m~\text{terms}\right]\)

  27. Find the last digit of the number \((32)^{32}.\)

  28. Prove that \(\sum_{r = 0}^k (-3)^{r - 1}.{}^{3n}C_{2r - 1} = 0,\) where \(k = \frac{3n}{2}\) and \(n\) is a positive even number.

  29. If \(t_0, t_1, t_2, t_3, \ldots\) be the terms of expansion \((a + x)^n,\) prove that \((t_0 - t_2 + t_4 - \ldots)^2 + (t_1 - t_3 + t_5 - \ldots)^2 = (a^2 + x^2)^n\)

If \((1 + x + x^2)^n = a_0 + a_1x + a_2x^2 + \ldots + a_{2n}x^{2n},\) show that

  1. \(a_0 + a_ 1 + a_2 + a_3 + \ldots + a_{2n} = 3^n\)

  2. \(a_0 - a_1 + a_2 - a_3 + \ldots + a_{2n} = 1\)

  3. \(a_0 + a_3 + a_6 + \ldots = 3^{2n - 1}\)

  4. If \(S_n = 1 + q + q^2 + \ldots + q^n\) and \(S_n^{'} = 1 + \left(\frac{q + 1}{2}\right) + \left(\frac{q + 1}{2}\right)^2 + \ldots + \left(\frac{q + 1}{2}\right)^n, q\neq 1,\) prove that \({}^{n + 1}C_1 + {}^{n + 1}C_2.S_1 + {}^{n + 1}C_3.S_2 + \ldots + {}^{n + 1}C_{n + 1}.S_n = 2^nS_n^{'}\)

  5. Find the number of rational terms in the expansion of \((\sqrt[4]{9} + \sqrt[6]{8})^{1000}\)

  6. Find the sum of rational terms in the expansion of \((\sqrt[3]{2} + \sqrt[5]{3})^{15}\)

  7. Determine the vlaue of \(x\) in the expansion of \((x + x\log_{10}x)^5\) if the third term in that expansion is \(1,000,000.\)

  8. Expand \(\left(x + 1 - \frac{1}{x}\right)^3\)

  9. Find the value of \(x\) for which the sixth term of \(\left(\sqrt{2^{\log(10 - 3^x)}} + \sqrt[5]{2^{(x - 2)\log 3}}\right)^m\) is equal to \(21\) and coefficients of seconds, third and fourth terms are the first, third and fifth terms of an A.P. Base of \(\log\) is \(10\).

  10. Find the values of \(x\) for which the sixth term of the expansion \(\left[2^{\log_2\sqrt{9^{x - 1} + 7}} + \frac{1}{2^{\frac{1}{5}\log_2(3^{x - 1} + 1)}}\right]^7\) is equal to 84.

  11. If \(n\) is a positive integer, prove that \(\frac{1}{(81)^n} - \frac{10}{(81)^n}{}^{2n}C_1 + \frac{10^2}{(81)^n}{}^{2n}C_2 - \frac{10^3}{(81)^n}{}^{2n}C_3 + \ldots + \frac{10^{2n}}{(81)^n} = 1\)

  12. Find the value of \(\lim_{n \to \infty}S_n = C_n - \frac{2}{3}C_{n - 1} + \left(\frac{2}{3}\right)^2C_{n - 2} - \ldots + (-1)^n\left(\frac{2}{3}\right)^nC_0\)

  13. If \(N = (6\sqrt{6} + 14)^{2n + 1}\) and \(F\) be the fractional part of \(N,\) prove that \(NF = 20^{2n + 1}\)

  14. Show that \(\sum_{r = 0}^n(-1)^r.{}^nC_r\left[\frac{1}{2^r} + \frac{3^r}{2^{2r}} + \frac{7^r}{w^{3r}} + \ldots~\text{up to}~n~\text{terms}\right] = \frac{1}{2^n - 1} - \frac{1}{2^{n^2}(2^n - 1)}\)

  15. Find the digits at units, tens and hundereds place in the number \((17)^256.\)

  16. Show that for \(n \geq 3, n^{n + 1} > (n + 1)^n, n\) is a positive integer.

  17. Show that \(2 < \left(1 + \frac{1}{n}\right)^n < 3\) for \(n \in N\)

  18. Show that \(1992^{1998} - 1955^{1998} - 1938^{1998} + 1901^{1998}\) is divisible by \(1998.\)

  19. Show that \(53^{53} - 33^{33}\) is divisible by \(10.\)

  20. Let \(k\) and \(n\) be positive integers and \(S_K = 1^k + 2^k + \ldots + n^k,\) show that \({}^{m + 1}C_1S_1 + {}^{m + 1}C_2S_2 + \ldots + {}^{m + 1}C_mS_m = (n + 1)^{m + 1} - n - 1\)

  21. Find \(\sum_{i = 1}^k\sum_{k = 1}^n{}^nC_k{}^kC_i, i\leq k\)

  22. Prove that \(\sum_{r = 0}^n(-1)^r.{}^nC_r \frac{1 + r\log_e 10}{(1 + \log_e10^n)^r} = 0\)

  23. Find the remainder when \(32^{32^{32}}\) is divided by \(7.\)

  24. If \(\sum_{r=0}^{2n}a_r(x - 2)^r = \sum_{r=0}^{2n}b_r(x - 3)^r\) and \(a_r = 1 \forall r \geq n,\) then show that \(b_n = {}^{2n + 1}C_{n + 1}\)

  25. Find the coefficient of \(x^{50}\) in \((1 + x)^{1000} + 2x(1 + x)^{999} + 3x^2(1 + x)^{998} + 1001x^{1000}\)

  26. Show that \({}^nC_n + {}^{n + 1}C_n + {}^{n + 2}C_n + \ldots + {}^{n + k}C_n = {}^{n + k + 1}C_{n + 1}\)

  27. Find the coefficient of \(x^n\) in \((1 + x + 2x^2 + 3x^3 + \ldots + nx^n)^2.\)

  28. Find the coefficient of \(x^k, 0\leq k\leq n\) in tthe expansion of \(1 + (1 + x) + (1 + x)^2 + \ldots + (1 + x)^n\)

  29. Find the coefficient of \(x^3\) in \((x + 1)^n + (x + 1)^{n - 1}(x + 2) + (x + 1)^{n - 2}(x + 2)^2 + \ldots + (x + 2)^n\)

  30. Simplify \(\left(\frac{a + 1}{a^{\frac{2}{3}} - a^{\frac{1}{3}} + 1} - \frac{a - 1}{a - a^{\frac{1}{2}}}\right)^{10}\) into a binomial and determine the term independent of \(a.\)

  31. Find the coefficient of \(x^2\) in \(\left(x + \frac{1}{x}\right)^{10}(1 - x + 2x^2)\)

  32. Find the coefficient of \(x^4\) in the expression of \((1 + x - 2x^2)^6\)

  33. Find the term indepndent of \(x\) in \((1 + x + 2x^3)\left(\frac{3}{2}x^2 - \frac{1}{3x}\right)^9\)

  34. Find the term independent of \(x\) in \(\left(x^2 + \frac{1}{x^3}\right)^7(2 - x)^{10}\)

  35. Find the term independent of \(x\) in \((1 + x + x^{-2} + x^{-3})^{10}\)

  36. Let \((1 + x^2)^2(1 + x)^n = \sum_{k = 0}^{n + 4}a_kx^k\)

    If \(a_1, a_2\) and \(a_3\) are in A.P., find \(n.\)

  37. Show that \({}^mC_1 + {}^{m + 1}C_2 + {}^{m + 2}C_3 + \ldots + {}^{m + n - 1}C_n = {}^nC_1 + {}^{n + 1}C_2 + {}^{n + 2}C_3 + \ldots + {}^{n + m - 1}C_n\)

  38. If \(n\in N\) and \((1 + x + x^2)^n = \sum_{r = 0}^{2n}a_rx^r,\) prove that

    1. \(a_r = a_{2n - r}\)

    2. \(a_0 + a_1 + a_2 + \ldots + a_{n - 1} = \frac{1}{2}(3^n - a_n)\)

    3. \((r + 1)a_{r + 1} = (n - r)a_r + (2n - r + 1)a-{r - 1},\) where \(0 < r < 2n\)

  39. If \((1 - x^3)^n = \sum_{r = 0}^n a_rx^r.(1 - x)^{3n - 2r},\) where \(n\ in N,\) then find \(a_r.\)

  40. Show that the coefficient of middle term in the expansion of \((1 + x)^2n\) is double the coefficient of \(x^n\) the expansion of \((1 + x)^{2n - 1}\)

  41. Find the value of \(r\) for which \({}^{200}C_r\) is greatest.

  42. Committees of how many persons should be made out of \(20\) persons so that number of committees is maximum.

  43. Show that the number of permutations which can be formed from \(2n\) letters which are either ‘a’ or ‘b’ is greatest when the number of a’s is equal to the number of b.

  44. Find the consecutive terms in the binomial expansion of \((3 + 2xy)^7\) whose coefficients are equal.

  45. Find the sum of coefficients in the expansion of \((1 + 5x^2 - 7x^3)^{2000}\)

  46. If the sum of the coefficients in the expansion of \(\left(3^{-\frac{x}{4}} + 3^{\frac{5x}{4}}\right)^n\) is \(64\) and the term with greatest coefficient exceeds the third term by \((n - 1)\) and \([\alpha] = x,\) where \([\alpha]\) denotes the integral part of \(\alpha\) find the value of \(\alpha.\)

  47. Find the sum of coefficients in the expansion of the binomial coefficients \((5p - 4q)^n,\) where \(n\) is a positive integer.

  48. Find the sum of the coefficients of the polynomial \((1 - 3x + x^3)^{201} . (1 + 5x - 5x^2)^{503}\)

  49. If the sum of the coefficients in the expansion of \((tx^2 - 2x + 1)^n\) is equal to the sum of coefficients in the expansion of \((x - ty)^n,\) where \(n\in N,\) then find the value of \(t.\)

  50. If \(a_0, a_1, a_2, \ldots, a_n\) be the successive coefficient of \((1 + x)^n,\) show that

    \((a_0 - a_2 + a_4 - \ldots)^2 + (a_1 -a_3 + a_5 - \ldots)^2 = a_0 + a_1 + \ldots + a_n = 2^n\)

  51. Find the greatest term in the expansion of \(\sqrt{3}\left(1 + \frac{1}{\sqrt{3}}\right)^{20}\)

  52. In the expansion \((x + a)^{15},\) if the eleventh term is the G.M. of the eightth and twelfth terms, which term in the expression is the greatest?

  53. If the greatest term in the expansion of \((1 + x)^{2n}\) has the greatest coefficient if and only if \(x \in \left(\frac{10}{11}, \frac{11}{10}\right)\) and the fourth term in the expansion of \(\left(kx + \frac{1}{x}\right)^m,\) is \(\frac{m}{4},\) then find the value of \(mk.\)

  54. Given that the \(4^{th}\) term in the exxpansion of \(\left(2 + \frac{3}{8}x\right)^{10}\) has the maximum numerical value, find the range of values of \(x\) for which this would be true.

  55. If \(n\) is a positive integer, show that \(9^n + 7\) is divisible by \(8.\)

  56. If \(n\) is a positive integer, show that \(3^{2n + 1} + 2^{n + 2}\) is divisible by \(7.\)

  57. Show that the roots of the equation \(ax^2 + 2bx + c = 0\) are real and unequal, where \(a, b, c\) are three consecutive coefficients in any binomial expansion with positive integral index.

  58. Show that no three consecutive binomial coefficients can be in G.P. or H.P.

  59. If \((1 + x)^n = C_0 + C_1x + C_2x^2 + \ldots + C_nx^n,\) show that \(C_n - 2.C_1 + 3.C_2 - \ldots +(-1)^n(n + 1)C_n = 0\)

  60. If \((1 + x)^n = C_0 + C_1x + C_2x^2 + \ldots + C_nx^n,\) show that \(C_0 -3.C_1 + 5.C_2 - \ldots + (-1)^n(2n + 1)C_n = 0\)

  61. If \((1 + x)^n = C_0 + C_1x + C_2x^2 + \ldots + C_nx^n,\) show that \(a - (a - 1)C_1 + (a - 2)C_2 - (a - 3)C_3 + \ldots + (-1)^n(a - n)C_n = 0\)

  62. Prove that \(\sum_{r = 0}^n r^2.{}^nC_r p^rq^{n - r} = npq + n^2p^2\) if \(p + q = 1\)

  63. If \(C_r\) stands for \({}^nC_r,\) show that \(2.C_0 + \frac{2^2}{2}C_1 + \frac{2^3}{3}C_2 + \ldots + \frac{2^{11}}{11}C_{11} = \frac{3^{11} - 1}{11}\)

  64. Prove that \(C_1 - \frac{1}{2}C_2 + \frac{1}{3}C_3 - \ldots + \frac{(1)^nC_n}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}\)

  65. Prove that \(\frac{C_0}{1} - \frac{C_1}{5} + \frac{C_2}{9} - \ldots + (-1)^n\frac{C_n}{4n + 1} = \frac{n.4^n}{1.5.9.\ldots (4n + 1)}\)

  66. Show that \(\frac{C_0}{n} - \frac{C_1}{n + 1} + \frac{C_2}{n + 2} - \ldots + (-1)^n\frac{C_n}{2n} = \frac{n!(n - 1)!}{(2n)!}\)

  67. Show that \(\frac{C_0}{n(n + 1)} - \frac{C_1}{(n + 1)(n + 2)} + \frac{C_2}{(n + 2)(n + 3)} - \ldots + (-1)^n\frac{C_{n}}{2n(2n + 1)} = \frac{1}{(2n + a)}.\frac{1}{{}^{2n}C_{n - 1}}\)

  68. Show that \(\frac{C_0}{x} - \frac{C_1}{x + 1} + \frac{C-2}{x + 2} - \ldots + (-1)^n\frac{C_n}{x + n} = \frac{n!}{x(x + 1)\ldots (x + n)}\)

  69. Show that \(C_0^2 - C_1^2 + C_2^2 - \ldots + (-1)^nC_n^2 = 0\) or \((-1)^{\frac{n}{2}}. \frac{n!}{\left(\frac{n}{2}\right)!}^2\) according as \(n\) is odd or even.

  70. Show that \({}^mC_r{}^nC_0 + {}^mC_{r - 1}{}^nC_1 + {}^mC_{r - 2}{}^nC_2 + \ldots + {}^mC_0{}^nC_r = {}^{m + n}C_r,\) where \(m, n, r\) are positive integers and \(r < m, r < n.\)

  71. \({}^{2n}C_0^2 - {}^{2n}C_1^2 + {}^{2n}C_2^2 - \ldots + (-1)^{2n}.{}^{2n}C_{2n}^2 = (-1)^n.{}^{2n}C_n.\)

  72. Show that \(C_1^2 + 2.C_2^2 + 3.C_3^2 + \ldots + n.C_n^2 = \frac{(2n - 1)!}{[(n - 1)!]^2}\)

  73. Show that \(C_0^2 + \frac{C_1^2}{2} + \frac{C_2^2}{3} + \ldots + \frac{C_n^2}{n + 1} = \frac{(2n + 1)!}{[(n + 1)!]^2}\)

  74. If \(n\) is a positive integer in \((1 + x)^n,\) show that \(2.\frac{\left(\frac{n}{2}!\right)^2}{n!}[c_0^2 -2.C_1^2 + 3.C_2^2 - \ldots + (-1)^n.(n + 1)C_n^2] = (-1)^{\frac{n}{2}}(2 + n)\)

  75. Show that \(\sum_{0\leq i \leq n}\sum_{0\leq i \leq j\leq n}C_i C_j = 2^{2n - 1}- \frac{(2n)!}{2(n!)^2}\)

  76. Show that \(\sum_{r=0}^nC_r^3\) is equal to the coefficient of \(x^ny^n\) in the expansion of \([(1 + x)(1 + y)(x + y)]^n.\)

  77. Let \(n\) be a positive integer and \((1 + x + x^2)^n = a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots + a_{2n}x^{2n},\) show that \(a_0^2 - a_1^2 + a_2^2 - \ldots + a_{2n}^2 = a_n\)

  78. Let \(n\) be a positive integer and \((1 + x + x^2)^n = a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots + a_{2n}x^{2n},\) show that \(a_0^2 - a_1^2 + a_2^2 - \ldots + (-1)^na_{n - 1}^2 = \frac{1}{2}a_n[1 - (-1)^na_n]\)

  79. Show that \(\sum_{0\leq i \leq n}\sum_{0\leq i \leq j\leq n} (C_i + C_j)^2 = (n - 1){}^{2n}C_n + 2^{2n}\)

  80. Show that \(\sum_{0\leq i \leq n}\sum_{0\leq i \leq j\leq n} (i + j)C_iC_j = n(2^{2n - 1} - \frac{1}{2}{}^{2n}C_n)\)

  81. Show that \((C_0 + C_1)(C_1 + C_2)(C_2 + C_3)\ldots (C_{n - 1} + C_n) = \frac{(n + 1)^n}{n!}C_1.C_2.\ldots .C_n\)

  82. If \(n\) be a positive integer prove that \(\frac{1}{1!(n - 1)!} + \frac{1}{3!(n - 3)!} + \frac{1}{5!(n - 5)! + \ldots + \frac{1}{(n - 1)!1!}} = \frac{2^{n - 1}}{n!}\)

  83. Prove that \(\frac{3!}{2(n + 3)} = \sum_{r=0}^n(-1)^r.\left(\frac{{}^nC_r}{{}^{r + 3}C_r}\right)\)

  84. Show that for \(m\geq 2, C_0 - C_1 + C_2 - \ldots + (-1)^{m - 1}C_{m - 1} = (-1)^{m - 1}\frac{(n- 1)(n - 2) \ldots (n - m + 1)}{(m - 1)!}\)

  85. Find the GCD of \({}^2nC_1, {}^{2n}C_3,{}^{2n}C_5, \ldots, {}^{2n}C_{2n - 1}\)

  86. Show that \(\sum_{r = 0}^n{}^nC_r \sin rx\cos (n - r)x = 2^{n - 1}\sin nx\)

  87. Show that \(C_1 -2.C_2 + 3.C_3 - \ldots + (-1)^{n - 1}n.C_n= 0\)

  88. \(a.C_0 + (a - b).C_1 + (a - 2b)C_2 + \ldots + (a - nb).C_n = 2^{n - 1}(2a - nb)\)

  89. \(a^2.C_0 - (a - 1)^2.C_1 + (a - 2)^2.C_2 - \ldots + (-1)^n(a - n)^2.C_n = 0; n > 3\)

  90. If \(a_0, a_1, a_2, \ldots, a_n\) be in an A.P, prove that \(a_0 - a_1.C_1 + a_2.C_2 - \ldots + (-1)^n a_n. C_n = 0\)

  91. Show that for \(n > 3, \sum_{r = 0}^n(-1)^r(a - r)(b - r)C_r = 0\)

  92. Show that for \(n > 3, \sum_{r = 0}^n(-1)^r(a - r)(b - r)(c - r)C_r = 0\)

  93. Find the value of \(n\) for \(\frac{C_0}{2^n} + \frac{2.C_1}{2^n} + \frac{3.C_2}{2^n} + \ldots + \frac{(n + 1)C_n}{2^n} = 16\) is true?

  94. If \(a_0, a_1, a_2, \ldots, a_{n + 1}\) be in an A.P, prove that \(\sum_{k = 0}^n a_{k + 1}C_k = 2^{n - 1}(a_1 + a_{n + 1})\)

  95. If \(s = \frac{n + 1}{2}[2a + nd]\) and \(S = a + (a + d)C_1 + (a + 2d)C_2 + \ldots + (a + nd)C_n,\) prove that \((n + 1)S = 2^n.s\)

  96. If \((1 + x + x^2 + \ldots + x^p)^n = a_0 + a_1x + a_2x^2 + \ldots + a_{np}x^{np},\) show that \(a + 2a_2 + 3a_3 + \ldots + npa_{np} = \frac{1}{2}np(p + 1)^n\)

  97. Prove that \(C_0 - 2^2C_1 + 3^2 C_2 - \ldots + (-1)^n(n + 1)^2C_n = 0, n > 2\)

  98. Show that \(\frac{C_1}{2} + \frac{C_3}{4} + \frac{C_5}{6} + \ldots = \frac{2^n - 1}{n + 1}\)

  99. Show that \(\frac{C_0}{1.2} - \frac{C_1}{2.3} + \frac{C_2}{3.4} + \ldots + (-1)^n\frac{C_n}{(n + 1)(n + 2)} = \frac{1}{n + 2}\)

  100. Show that \(\frac{C_0}{2} - \frac{C_1}{3} + \frac{C-2}{4} - \ldots + (-1)^n\frac{C_n}{n + 2} = \frac{1}{(n + 1)(n + 2)}\)

  101. Show that \(\frac{C_0}{3} - \frac{C_1}{4} + \frac{C-2}{5} - \ldots + (-1)^n\frac{C_n}{n + 3} = \frac{2}{(n + 1)(n + 2)(n + 3)}\)

  102. Show that \(3.C_0 + 3^2\frac{C_1}{2} + 3^3\frac{C_2}{3} + \ldots + 3^{n + 1}\frac{C_n}{n + 1} = \frac{4^{n + 1} - 1}{n + 1}\)

  103. Show that \(\sum_{k = 0}^{15}\frac{{}^{15}C_k}{(k + 1)(k + 2)} = \frac{26{17} - 18}{16.17}\)

  104. Show that \(\frac{C_0}{1} - \frac{C_1}{4} + \frac{C_2}{7} - \ldots + (-1)^n\frac{C_n}{3n + 1} = \frac{3^nn!}{1.4.7.\ldots + (3n + 1)}\)

  105. Show that \(\sum_{r = 0}^n \frac{(-1)^rC_r}{(r + 1)(r + 2)} = \frac{1}{n + 2}\)

  106. Prove that \(\sum_{r = 0}^n \frac{C_re^{r + 3}}{(r + 1)(r + 2)(r + 3)} = \frac{4^{n + 3} - 1 - \frac{3}{2}(n + 3)(en + 8)}{(n + 1)(n + 2)(n + 3)}\)

  107. Prove that \(\sum_{r = 0}^n \frac{r + 2}{r + 1}C_r = \frac{2^n(n + 3) - 1}{n + 1}\)

  108. Show that \(\sum_{r = 0}^n \frac{3^{r + 4}C_r}{(r + 1)(r + 2)(r + 3)(r + 4)} = \frac{1}{(n + 1)(n + 2)(n + 3)(n + 4)}\left[4^{n + 4} - sum_{k = 0}^3 {}^{n + 4}C_k3^k\right]\)

  109. Show that \(\sum_{r=0}^{n - 3}C_rC_{r + 3} = \frac{(2n)!}{(n + 3)!(n - 3)!}\)

  110. Show that sum of the product taken two at a time from \(C_0, C_1, C_2, \ldots\) is \(2^{2n - 1}\frac{(2n - 1)!}{n!(n - 1)!}\)

  111. If \(S_n = C_0C_1 + C_1C_2 + \ldots + C_{n - 1}C_n\) and \(\frac{S_{n + 1}}{S_n} = \frac{15}{4},\) find n.

  112. Show that \(C_0^2 + \frac{C_1^2}{2} + \frac{C_2^2}{3} + \ldots + \frac{C_n^2}{n + 1} = \frac{(n + 2)(2n - 1)!}{n!(n - 1)!}\)

  113. Show that \(C_1^2 -2.C_2^2 + 3.C_3^2 - \ldots - 2nC_{2n}^2 = (-1)^{n - 1}.n.C_n\) where \(C_r = {}^{2n}C_r\)

  114. Show that \({}^{25}C_0^2 - {}^{25}C_1^2 + {}^{25}C_2^2 - \ldots - {}^{25}C_25^2 = 0\)

  115. Show that \(C_0.{}^{2n}C_n - C_1{}^{2n - 2}C_n + C_2{}^{2n -4}C_n - \ldots = 2^n\)

  116. Show that \(\sum_{0\leq i \leq n}\sum_{0\leq i \leq j\leq n}(i + j)(C_i + C_j + C_iC_j) = n^2.2^n + n\left(2^{2n - 1} - \frac{(2n)!}{2(n!)^2}\right)\)

  117. If \((1 + x + x^2)^n = a_0 + a_1x + a_2x^2 + \ldots + a_{2n}x^{2n}\) show that \(a_0a_{2r} - a_1a_{2r + 1} + a_2a_{2r + 2} - \ldots + a_{2n -2r}a_{2n} = a_{n + r}\)

  118. If \(a_r = \frac{1.3.5.\ldots (2r - 1)}{2.4.6.\ldots 2r}\) then show that \(a_{2n + 1} + a_1a_{2n} + a_2a_{2n - 1} + \ldots + a_na_{n + 1} = \frac{1}{2}\)

  119. If \(P_n\) denotes the product of all coefficients in the expansion of \((1 + x)^n,\) show that \(\frac{P_{n + 1}}{P_n} = \frac{(n + 1)^n}{n!}\)

  120. Show that \(\sum_{r = 1}^nr^3\left(\frac{C_r}{C_{r - 1}}\right)^2 = \frac{1}{12}n(n + 1)^2(n + 2)\)

  121. Show that \(C_3 + C_7 + C_11 + \ldots = \frac{1}{3}\left[2^{n - 1} -2^{\frac{n}{2}}\sin \frac{n\pi}{4}\right]\)

  122. If \((1 + x + x^2)^{20} = a_0 + a_1x + a_2x^2 + \ldots + a_{40}x^{40},\) then find the value of \(a_0 + a_2 + a_4 + \ldots + a_{40}\)

  123. If \((1 + x + x^2)^{20} = a_0 + a_1x + a_2x^2 + \ldots + a_{40}x^{40},\) then find the value of \(a_1 + a_3 + a_5 + \ldots + a_{39}\)

  124. Show that \(C_1 - \frac{C_2}{2} + \frac{C_3}{3} - \ldots + (-1)^n\frac{C_n}{n} + \frac{1}{n(n - 1)} + \frac{2}{(n - 1)(n - 2)} + \ldots + \frac{n - 2}{2.3} = \frac{n + 1}{2}\)

  125. Show that \(\sum_{0\leq i \leq n}\sum_{0\leq i \leq j\leq n} \frac{i}{C_i + \frac{j}{C_j}} = \frac{n^2}{2}\sum_{r = 0}^n \frac{1}{C_r}\)

  126. Show that \(\sum_{0\leq i \leq n}\sum_{0\leq i \leq j\leq n} i.j.C_i.C_j = n^2\left[2^{2n - 3} - \frac{1}{2}{}^{2n - 2}C_{n - 1}\right]\)

  127. Prove that \(C_1 - \left(1 + \frac{1}{2}\right)C_2 + \left(1 + \frac{1}{2} + \frac{1}{3}\right)C_3 - \ldots + (-1)^n\left(1 + \frac{1}{2} + \ldots + \frac{1}{n}\right)C)n = \frac{1}{n}\)

  128. Find the coefficient of \(x^5\) in the expansion of \((1 + 2x + 3x^2)^4\)

  129. Find the coefficient of \(x^3y^4z^2\) in the expansion of \((2x - 3y + 4z)^9\)

  130. Find the number of terms in \((2x - 3y + 4z)^{100}\)

  131. Find the coefficient of \(x^4\) in the expansion of \((1 + x + x^2)^3\)

  132. Find the coefficient of \(x^{10}\) in \((1 + x + x^2 + x^3 + x^4 + x^5)^3\)

  133. Find the coefficient of \(x^7\) in \((1 + 3x - 2x^3)^{10}\)

  134. Find the coefficient of \(x^3y^4z^5\) in \((xy + yz + zx)^6\)

  135. Find the greatest coefficient in \((w + x + y + z)^{15}\)

  136. Find the number of terms in \((a + b + c + d + e)^{100}\)

  137. If \(|x| < 1,\) show that \((1 + x)^{-2} = 1 + 2x + 3x^2 + 4x^3 + \ldots ~\text{to}~\infty\)

  138. Find \(a, b\) so that the coefficient of \(x^n\) in the expansionof \(\frac{(a + bx)}{(1 - x)^2}\) may be \(2n + 1\) and hence find the sum of the series \(1 + 3\left(\frac{1}{2}\right) + 5\left(\frac{1}{2}\right)^2 + \ldots\)

  139. Sum the series \(1 + \frac{1}{3} + \frac{1.3}{3.6} + \frac{1.3.5}{3.6.9} + \ldots~\text{to}~\infty\)

  140. If \(|x| <1,\) show that \((1 - x)^{-1} = 1 + x + x^2 + x^3 + \ldots~\text{to}~\infty\)

  141. If \(|x| <1,\) show that \((1 + x)^{-1} = 1 - x + x^2 - x^3 + \ldots~\text{to}~\infty\)

  142. If \(|x| <1,\) show that \((1 + x)^{-2} = 1 - 2x + 3x^2 - 4x^3 + \ldots~\text{to}~\infty\)

  143. If \(|x| <1,\) show that \((1 - x)^{-3} = 1 + 3x + 6x^2 + 10x^3 + \ldots~\text{to}~\infty\)

  144. If \(|x| <1,\) show that \((1 + x)^{-3} = 1 - 3x + 6x^2 - 10x^3 + \ldots~\text{to}~\infty\)

  1. If \(|x| <1,\) show that \((1 + x)^{\frac{1}{5}} = 1 - \frac{x}{5} + \frac{3x^2}{25} - \frac{11x^3}{125} + \ldots~\text{to}~\infty\)

  2. Find the first four terms of \(\left(\frac{2x}{3} - \frac{3}{2x}\right)^{-\frac{3}{2}}\)

  3. Find the first three terms of \(\left(1 - \frac{x}{2}\right)^{-2}\)

  4. Find the coefficient of \(x^6\) in \((1 - 2x)^{-\frac{5}{2}}\)

  5. Find the \((r + 1)^{th}\) term and its coefficient in \((1 - 2x)^{-\frac{1}{2}}\)

  6. Find the cube root of \(1001\) correct to four places of decimal.

  7. Show that \((1 + 2x + 3x^2 + 4x^3 + \ldots~\text{to}~\infty)^\frac{3}{2} = 1 + 3x + 6x^2 + 10x^3 + \ldots~\text{to}~\infty,\) where \(|x| < 1\)

  8. Sum the series \(1 + \frac{1}{4} + \frac{1.3}{4.8} + \frac{1.3.5}{4.8.12} + \ldots~\text{to}~\infty\)

  9. Sum the series \(1 + \frac{2}{6} + \frac{2.5}{6.12} + \frac{2.5.8}{6.12.18} + \ldots~\text{to}~\infty\)

  10. If \(y = x - x^2 + x^3 - x^4 + \ldots~\text{to}~\infty,\) show that \(x = y + y^2 + y^3 + \ldots~\text{to}~\infty\)

  11. Show that the coefficient of \(x^n\) in \((1 + x + x^2)^{-1}\) is \(1, 0, -1\) as \(n\) is of the form \(3m, 3m -1, 3m + 1\)

  12. Show that \(\frac{1}{e} = 2\left[\frac{1}{3!} + \frac{2}{5!} + \frac{3}{7!} + \ldots~\text{to}~\infty\right]\)

  13. Sum the series \(1 + \frac{2^2}{2!} + \frac{3^2}{3!} + \frac{4^2}{4!} + \ldots~\text{to}~\infty\)

  14. Show that \(\log 2 = \frac{1}{1.2} + \frac{1}{3.4} + \frac{1}{5.6} + \ldots~\text{to}~\infty\)

  15. If \(y = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots~\text{to}~\infty,\) show that \(x = y + \frac{y^2}{2!} + \frac{y^3}{3!} + \ldots~\text{to}~\infty\)

  16. If \(\alpha, \beta\) be the roots of the equation \(ax^2 + bx + c = 0,\) show that \(\log(a - bx + cx^2) = =log a + (\alpha + \beta)x - \frac{(\alpha^2 + \beta^2)}{2}x^2 + \ldots~\text{to}~\infty\)

  17. Prove that \(\frac{e^x - 1}{x} = 1 + \frac{x}{2!} + \frac{x^2}{3!} + \ldots~\text{to}~\infty\)

  18. Prove that \(\frac{e^x - e^{-x}}{x} = 2\left[1 + \frac{x^2}{3!} + \frac{x^4}{5!} + \ldots~\text{to}~\infty\right]\)

  19. Prove that \(\frac{a^x - 1}{x} = \log a + \frac{x(\log a)^2}{2!} + \frac{x^2(\log a)^3}{3!} + \ldots~\text{to}~\infty\)

  20. Sum the series \(\frac{1}{3!} + \frac{2}{5!} + \frac{3}{7!} + \ldots~\text{to}~\infty\)

  21. Sum the series \(\frac{1}{2!} + \frac{3}{4!} + \frac{5}{6!}+ \ldots~\text{to}~\infty\)

  22. Sum the seeries \(\frac{1}{2!} + \frac{1 + 2}{3!} + \frac{1 + 2 + 3}{4!} + \ldots~\text{to}~\infty\)

  23. Sum the series \(\frac{1^3}{1!} + \frac{2^3}{2!} + \frac{3^3}{3!} + \ldots~\text{to}~\infty\)

  24. Prove that \(1 -\log 2 = \frac{1}{2.3} + \frac{1}{4.5} + \frac{1}{6.7} + \ldots~\text{to}~\infty\)

  25. Prove that \(\log (1 + x) - \log(x - 1) = 2\left[\frac{1}{x} + \frac{1}{3x^3} + \frac{1}{5x^5} + \ldots~\text{to}~\infty\right]\)

  26. Prove that \(2\log x - \log(x + 1) - \log(x -1) = \frac{1}{x^2} + \frac{1}{2x^4} + \frac{1}{3x^6} + \ldots~\text{to}~\infty\)

  27. Prove that \(\log(1 + x)^{1 + x}\log(1 - x)^{1 - x} = 2.\left[\frac{x^2}{1.2} + \frac{x^4}{3.4} + \frac{x^6}{5.6} + \ldots~\text{to}~\infty\right]\)

  28. If \(\alpha, \beta\) be the roots of the equation \(x^2 -px + q = 0,\) show that \(\log(1 + px + qx^2) = (\alpha +\beta)x - \frac{\alpha^2 + \beta^2}{2}x^2 + \frac{\alpha^3 + \beta^3}{3}x^3 + \ldots~\text{to}~\infty\)