AM, GM, HM Relation and Problems ******************************** Arithmetic Means(A. M.) ======================== **Single A. M.** a number :math:`A` is said to be the single A. M. between two given numbers :math:`a` and :math:`b` if :math:`a, A, b` are in A. P. **Example:** Since :math:`1, 3, 5` are in A. P., therefore, :math:`3` is the single A. M., between :math:`1` and :math:`5`. **Arithmetic Means:** numbers :math:`A_1, A_2, ..., A_n` are said to be the n :math:`A. M.` between two given numbers :math:`a` and :math:`b` if :math:`a, A_1, A_2, ..., A_n, b` are in A. P. **Example:** Since :math:`1, 3, 5, 7, 9, 11` are in A. P., therefore :math:`3, 5, 7, 9` are the four A. M. between :math:`1` and :math:`11`. Single Arithmetic Mean between two given quantities --------------------------------------------------- Let :math:`a` and :math:`b` be the two given quantities and :math:`A` be the A. M. between them. Then, :math:`a, A, b` will be in A. P. :math:`\therefore A - a = b - A` or :math:`A = \frac{a + b}{2}` To insert :math:`n` A. M. between two given quantities ------------------------------------------------------ Let :math:`A_1, A_2, ..., A_n` be the :math:`n` A. M. between two given quantities :math:`a` and :math:`b`. Then, :math:`a, A_1, A_2, ..., A_n, b` will be in A. P. Now, :math:`b = a + (n + 2 - 1)d` where :math:`d` is common difference of A. P. :math:`d = \frac{b - 1}{n + 1}` Now, first A. M. :math:`A_1 = a + d = \frac{an + b}{n + 1}` Second A. M. :math:`A_2 = a + 2d = \frac{a(n - 1) + 2b}{n + 1}` ... nth A. A. :math:`A_n = a + nd = \frac{a + nb}{n + 1}` Geometric Means(G. M.) ====================== **Single Geometric Mean:** Single G. M. between two positive numbers :math:`a` abd :math:`b` is the positive square root of :math:`ab` **Example:** Since :math:`1, 3, 9` are in G. P., therefore :math:`3` is the geometric mean between :math:`1` and :math:`9`. **Geometric Means:** :math:`n` numbers :math:`G_1, G_2, ..., G_n` are said to be the G.M.'s between two given numbers :math:`a` and :math:`b` if :math:`a, G_1, G_2, ..., G_n, b` are in G. P. **Example:** Since :math:`1, 2, 4, 8, 16` are in G. P., therefore :math:`2, 4, 8` are the three G.M.'s between :math:`1` and :math:`16`. Single geometric mean between two given quantities -------------------------------------------------- Let :math:`a` and :math:`b` be the two positive numbers and :math:`G` be the single G. M. between them. Then :math:`a, G, b` will be in G. P. :math:`\therefore \frac{G}{a} = \frac{b}{G}` or :math:`G^2 = ab \therefore G = \sqrt{ab} [\because G > 0]` To insert :math:`n` G. M. between two given quantities -------------------------------------------------------- Let :math:`G_1, G2, ..., G_n` be the :math:`n` G. M. ebtweeb two given quantities :math:`a` and :math:`b`. Then :math:`a, G_1, G2, ..., G_n, b` will be in G. P. Now, :math:`b = (n+2)` th term of G. P. = :math:`ar^{n+1}`, where :math:`r` = common ratio of the G. P. :math:`\therefore r^{n + 1} = \frac{b}{a}` or :math:`r = \left(\frac{b}{a}\right)^\frac{1}{n + 1}` Now, first G. M. :math:`G_1 = ar = a\left(\frac{b}{a}\right)^\frac{1}{n + 1}` Second G. M. :math:`G_2 = ar^2 = a\left(\frac{b}{a}\right)^\frac{2}{n + 1}` ... :math:`n` th G. M. :math:`G_n = ar^n = a\left(\frac{b}{a}\right)^\frac{n}{n + 1}` Harmonic Means(H. M.) ===================== Numbers :math:`H_1, H_2, ..., H_n` are said to be the :math:`n` H.M. between two numbers :math:`a` and :math:`b` if :math:`a, H_1, H_2, ..., H_n, b` are in H. P. For example, :math:`\frac{1}{2}, \frac{1}{3}, \frac{1}{4}` are the H. M. between :math:`1` and :math:`\frac{1}{5}` because :math:`1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \frac{1}{5}` are in H. P. Single Harmonic Mean between two given quantities ------------------------------------------------- Let :math:`a` and :math:`b` be the two given quantities and :math:`H` be the H. M. between them. Then :math:`a, H, b` will be in H. P. :math:`\therefore \frac{1}{a}, \frac{1}{H}, \frac{1}{b}` will be in A. P. :math:`\therefore \frac{1}{H} - \frac{1}{a} = \frac{1}{b} - \frac{1}{H}` or :math:`\frac{2}{H} = \frac{1}{a} + \frac{1}{b}` or :math:`\frac{2}{H} = \frac{b + a}{ab} \therefore H = \frac{2ab}{a + b}` Hence, H. M. of :math:`a` and :math:`b = \frac{2ab}{a + b}` :math:`n` harmonic means between two given quantities ----------------------------------------------------- Let :math:`H_1, H_2, ..., H_n` be the :math:`n` H. M. between two given quantities :math:`a` and :math:`b` and :math:`d` be the c.d. of the corresponding A. P. Then :math:`a, H_1, H_2, ..., H_n, b` will be in H. P. :math:`\therefore \frac{1}{a}, \frac{1}{H_1}, \frac{1}{H_2}, ..., \frac{1}{H_n}, \frac{1}{b}` will be in A. P. Now :math:`\frac{1}{b} = (n + 2)` th term of A. P. :math:`= \frac{1}{a} + (n + 2 - 1)d` :math:`\therefore d = \frac{\frac{1}{b} - \frac{1}{a}}{n + 1} = \frac{a - b}{ab(n + 1)}` Now, :math:`\frac{1}{H_1} = \frac{1}{a} + d = \frac{bn + a}{ab(n + 1)}` :math:`\therefore H_1 = \frac{ab(n + 1)}{bn + a}` :math:`\frac{1}{H_2} = \frac{1}{a} + 2d = \frac{1}{a} + \frac{2(a - b)}{ab(n + 1)}` :math:`H_2 = \frac{ab(n + 1)}{2a + (n - 1)b}` ... :math:`\frac{1}{H_n} = \frac{1}{a + nd} = \frac{1}{a} + \frac{n(a - b)}{ab(n + 1)}` :math:`H_n = \frac{ab(n + 1)}{na + b}` Relation between A.M, G. M. and H. M. between two real and unequal quantities ----------------------------------------------------------------------------- Let :math:`a` and :math:`b` be two real, positive and unequal quantities and :math:`A, G` and :math:`H` be the single A. M., G. M. and H. M. respectively. Then, :math:`A = \frac{a + b}{2}, G = \sqrt{ab}, H = \frac{2ab}{a + b}` Now, :math:`AH = \frac{a + b}{2}.\frac{2ab}{a + b} = ab = G^2 \therefore \frac{G}{A} = \frac{H}{G}` Hence, :math:`A, G` and :math:`H` are in G. P. Again. :math:`A - G = \frac{a + b}{2} - \sqrt{ab} = \frac{a + b - 2\sqrt{ab}}{2}` :math:`= \frac{(\sqrt{a} - \sqrt{b})^2}{2} > 0~[\because a\ne 0]` Thus, :math:`A - G > 0` or :math:`A > G` Since, :math:`\frac{H}{G} = \frac{G}{A}` but :math:`\frac{G}{A} < 1 \therefore \frac{H}{G} < 1` Thus, :math:`A > G > H` For equal :math:`a` and :math:`b` it can be easily verified that :math:`A = G = H` Problems ======== 1. If :math:`n` arithmetic means are inserted between :math:`20` and :math:`80` such that first mean : last mean = 1 : 3. Find :math:`n`. 2. Prove that the sum of :math:`n` arithmetic means between two given numbers is :math:`n` times the single arithmetic mean between them. 3. Between two numbers whose sum is :math:`\frac{13}{6}`, an even number of arithmetic means are inserted. If the sum of means exceeds their number by unity find the number of means. 4. For what value of :math:`n, \frac{a^{n + 1} + b^{n + 1}}{a^n + b^n}, a\ne b` is the A. M. of :math:`a` and :math:`b`. 5. Insert :math:`4` G. M. between :math:`5` and :math:`160`. 6. Show that the product of :math:`n` geometric means inserted between two positive quantities is equal to the :math:`n` the power of the single geometric means inserted between them. 7. Insert :math:`6` harmonic means between :math:`3` and :math:`\frac{6}{23}`. 8. If the A. M. and G. M. between two numbers be :math:`5` and :math:`3` respectively. find the numbers. 9. If the A. M. between two numbers be twice their G. M. show that the ratio of the numbers is :math:`2 + \sqrt{3}: 2 - \sqrt{3}`. 10. If :math:`a` be one A. M. and :math:`g_1` abd :math:`g_2` be two G. M. between :math:`b` and :math:`c`, prove that :math:`g_1^{3} + g_2^3 = 2abc` 11. If :math:`a, b, c` be in G. P. and :math:`x, y` be the A. M. between :math:`a, b` and :math:`b, c` respectively, show that :math:`\frac{a}{x} + \frac{b}{y} = 2, \frac{1}{x} + \frac{1}{y} = \frac{2}{b}` 12. If :math:`A` be the A. M. and :math:`H` be the H. M. between two quantities :math:`a` and :math:`b`, prove that :math:`\frac{a - A}{a - H}.\frac{b - A}{b - H} = \frac{A}{H}` 13. If :math:`A_1, A_2` be the A. M., :math:`G_1, G_2` be the G. M. and :math:`H_1, H_2` be the H. M. between any two quantities, show that :math:`\frac{G_1G_2}{H_1H_2} = \frac{A_1 + A_2}{H_1 + H_2}` 14. The arithmetic mean of two numbers exceed their geometric mean by :math:`\frac{3}{2}` and the geometric mean exceeds their harmonic mean by :math:`\frac{6}{5}`, find the numbers. 15. If :math:`a, b, c, d` be four distinct quantities in H. P., show that (i) :math:`a + d > b + c` (ii) :math:`ad > bc` 16. If three positive uneuqal quantities :math:`a, b, c` be in H. P. prove that :math:`a^n + c^n > 2b^n`, where :math:`n` is a positive integer. 17. If :math:`x + y + z = 15` if :math:`a, x, y, z, b` are in A. P. and :math:`\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{5}{3}` if :math:`a, x, y, z, b` are in H. P. find :math:`a` and :math:`b`. 18. If :math:`x > 0`, prove that :math:`x + \frac{1}{x} \geq 2` 19. Insert :math:`8` A. M. between :math:`5` and :math:`32`. 20. Insert :math:`7` A. M. between :math:`2` and :math:`34`. 21. Insert :math:`17` A. M. between :math:`\frac{7}{2}` and :math:`-\frac{83}{2}`. 22. Between :math:`1` and :math:`31, n` A. M. are inserted such that ratio of the :math:`7` th and :math:`(n - 1)` th means is :math:`5:9`, find :math:`n`. 23. Find the relation between :math:`x` and :math:`y` in order that :math:`r` th mean between :math:`x` and :math:`2y` may be the same as :math:`r` th mean between :math:`2x` and :math:`y`; if :math:`n` arithmetic means are inserted in each case. 24. Insert :math:`7` geometric means between :math:`2` and :math:`162`. 25. Insert :math:`6` geometric means between :math:`\frac{8}{27}` and :math:`\frac{-81}{16}`. 26. If odd numbers of geometric means are inserted between two given quantities :math:`a` and :math:`b`, show that the middle geometric mean is :math:`\sqrt{ab}`. 27. Insert four harmonic means between :math:`1` and :math:`\frac{1}{11}`. 28. :math:`n` harmonic means have been inserted between :math:`1` and :math:`4` such that first mean:last mean = :math:`1:3`, find :math:`n`. 29. Find :math:`n` such :math:`\frac{a^{n+1}+b^{n+1}}{a^n + b^n}` may be a single harmonic mean between :math:`a` and :math:`b`. 30. If :math:`H_1, H_2, ..., H_n` be :math:`n` harmonic means between :math:`a` and :math:`b`, prove that :math:`\frac{H_1 + a}{H_1 - a} + \frac{H_n + b}{H_n - b} = 2n` 31. If :math:`A` be the A. M. and :math:`G` be the G. M. between two numbers, show that the numbers are :math:`A + \sqrt{A^2 - G^2}` and :math:`A - \sqrt{A^2 - G^2}` 32. If the ratio of A. M. and G. M. between two numbers :math:`a` and :math:`b` is :math:`m:n`, prove that :math:`a:b = m + \sqrt{m^2 - n^2}:m - \sqrt{m^2 - n^2}` 33. If one G. M. :math:`G` and two A. M. :math:`p` and :math:`q` be inserted between two given quantities, prove that :math:`G^2 = (2p - q)(2q - p)` 34. If one A. M. :math:`A` and two G. M. :math:`p` and :math:`q` be inserted between two numbers, show that :math:`\frac{p^2}{q} + \frac{q^2}{p} = 2A` 35. If A. M. between :math:`a` and :math:`b` is equal to :math:`m` times the H. M., prove that :math:`a:b = \sqrt{m}+\sqrt{m - 1}:\sqrt{m}-\sqrt{m - 1}` 36. If :math:`9` arithmetic means and :math:`9` harmonic means be inserted between :math:`2` and :math:`3`, prove that :math:`A + \frac{6}{H} = 5`, where :math:`A` is any arithmetic mean and :math:`H`, the corresponding harmonic mean. 37. If :math:`a` is the A. M. between :math:`b` and :math:`c`, :math:`b` the G. M. between :math:`a` and :math:`c`, then show that :math:`c` is the H. M. between :math:`a` and :math:`b`. 38. If :math:`a_1, a_2` be the two A. M., :math:`g_1, g_2` be the two G. M. and :math:`h_1, h_2` be the two H. M. between any two numbers :math:`x` and :math:`y`, show that :math:`a_1h_2 = a_2h_1 = g_1g_2 = xy` 39. If between any two quantities, there be inserted :math:`2n - 1` arithmetic, geometric and harmonic means, show that :math:`n` th means inserted are in G. P. 40. The A. M. between two numbers exceed their G. M. by :math:`2` and the G. M. exceeds the H. M. by :math:`\frac{8}{5}`, find the numbers. 41. The harmonic mean of two numbers is :math:`4`, their A. M. :math:`A` and G. M. :math:`G` satisfy the relation :math:`2A + G^2 =27`. Find the numbers. 42. If :math:`a, b, c` are in H. P., prove that i. :math:`a^2 + c^2 > 2b^2` ii. :math:`a^5 + c^5 > 2b^5` 43. Prove that :math:`b^2 > = < ac` according as :math:`a, b, c` are in A. P., G. P. and H. P.