36. AM, GM, HM Relation and Problems

36.1. Arithmetic Means(A. M.)

Single A. M. a number \(A\) is said to be the single A. M. between two given numbers \(a\) and \(b\) if \(a, A, b\) are in A. P.

Example: Since \(1, 3, 5\) are in A. P., therefore, \(3\) is the single A. M., between \(1\) and \(5\).

Arithmetic Means: numbers \(A_1, A_2, ..., A_n\) are said to be the n \(A. M.\) between two given numbers \(a\) and \(b\) if

\(a, A_1, A_2, ..., A_n, b\) are in A. P.

Example: Since \(1, 3, 5, 7, 9, 11\) are in A. P., therefore \(3, 5, 7, 9\) are the four A. M. between \(1\) and \(11\).

36.1.1. Single Arithmetic Mean between two given quantities

Let \(a\) and \(b\) be the two given quantities and \(A\) be the A. M. between them. Then, \(a, A, b\) will be in A. P.

\(\therefore A - a = b - A\) or \(A = \frac{a + b}{2}\)

36.1.2. To insert \(n\) A. M. between two given quantities

Let \(A_1, A_2, ..., A_n\) be the \(n\) A. M. between two given quantities \(a\) and \(b\).

Then, \(a, A_1, A_2, ..., A_n, b\) will be in A. P.

Now, \(b = a + (n + 2 - 1)d\) where \(d\) is common difference of A. P.

\(d = \frac{b - 1}{n + 1}\)

Now, first A. M. \(A_1 = a + d = \frac{an + b}{n + 1}\)

Second A. M. \(A_2 = a + 2d = \frac{a(n - 1) + 2b}{n + 1}\)

nth A. A. \(A_n = a + nd = \frac{a + nb}{n + 1}\)

36.2. Geometric Means(G. M.)

Single Geometric Mean: Single G. M. between two positive numbers \(a\) abd \(b\) is the positive square root of \(ab\)

Example: Since \(1, 3, 9\) are in G. P., therefore \(3\) is the geometric mean between \(1\) and \(9\).

Geometric Means: \(n\) numbers \(G_1, G_2, ..., G_n\) are said to be the G.M.’s between two given numbers \(a\) and \(b\) if

\(a, G_1, G_2, ..., G_n, b\) are in G. P.

Example: Since \(1, 2, 4, 8, 16\) are in G. P., therefore \(2, 4, 8\) are the three G.M.’s between \(1\) and \(16\).

36.2.1. Single geometric mean between two given quantities

Let \(a\) and \(b\) be the two positive numbers and \(G\) be the single G. M. between them.

Then \(a, G, b\) will be in G. P.

\(\therefore \frac{G}{a} = \frac{b}{G}\) or \(G^2 = ab \therefore G = \sqrt{ab} [\because G > 0]\)

36.2.2. To insert \(n\) G. M. between two given quantities

Let \(G_1, G2, ..., G_n\) be the \(n\) G. M. ebtweeb two given quantities \(a\) and \(b\).

Then \(a, G_1, G2, ..., G_n, b\) will be in G. P.

Now, \(b = (n+2)\) th term of G. P.

= \(ar^{n+1}\), where \(r\) = common ratio of the G. P.

\(\therefore r^{n + 1} = \frac{b}{a}\) or \(r = \left(\frac{b}{a}\right)^\frac{1}{n + 1}\)

Now, first G. M. \(G_1 = ar = a\left(\frac{b}{a}\right)^\frac{1}{n + 1}\)

Second G. M. \(G_2 = ar^2 = a\left(\frac{b}{a}\right)^\frac{2}{n + 1}\)

\(n\) th G. M. \(G_n = ar^n = a\left(\frac{b}{a}\right)^\frac{n}{n + 1}\)

36.3. Harmonic Means(H. M.)

Numbers \(H_1, H_2, ..., H_n\) are said to be the \(n\) H.M. between two numbers \(a\) and \(b\) if \(a, H_1, H_2, ..., H_n, b\) are in H. P. For example, \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\) are the H. M. between \(1\) and \(\frac{1}{5}\) because \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \frac{1}{5}\) are in H. P.

36.3.1. Single Harmonic Mean between two given quantities

Let \(a\) and \(b\) be the two given quantities and \(H\) be the H. M. between them. Then \(a, H, b\) will be in H. P.

\(\therefore \frac{1}{a}, \frac{1}{H}, \frac{1}{b}\) will be in A. P.

\(\therefore \frac{1}{H} - \frac{1}{a} = \frac{1}{b} - \frac{1}{H}\) or \(\frac{2}{H} = \frac{1}{a} + \frac{1}{b}\) or

\(\frac{2}{H} = \frac{b + a}{ab} \therefore H = \frac{2ab}{a + b}\)

Hence, H. M. of \(a\) and \(b = \frac{2ab}{a + b}\)

36.3.2. \(n\) harmonic means between two given quantities

Let \(H_1, H_2, ..., H_n\) be the \(n\) H. M. between two given quantities \(a\) and \(b\) and \(d\) be the c.d. of the corresponding A. P.

Then \(a, H_1, H_2, ..., H_n, b\) will be in H. P.

\(\therefore \frac{1}{a}, \frac{1}{H_1}, \frac{1}{H_2}, ..., \frac{1}{H_n}, \frac{1}{b}\) will be in A. P.

Now \(\frac{1}{b} = (n + 2)\) th term of A. P.

\(= \frac{1}{a} + (n + 2 - 1)d\)

\(\therefore d = \frac{\frac{1}{b} - \frac{1}{a}}{n + 1} = \frac{a - b}{ab(n + 1)}\)

Now, \(\frac{1}{H_1} = \frac{1}{a} + d = \frac{bn + a}{ab(n + 1)}\)

\(\therefore H_1 = \frac{ab(n + 1)}{bn + a}\)

\(\frac{1}{H_2} = \frac{1}{a} + 2d = \frac{1}{a} + \frac{2(a - b)}{ab(n + 1)}\)

\(H_2 = \frac{ab(n + 1)}{2a + (n - 1)b}\)

\(\frac{1}{H_n} = \frac{1}{a + nd} = \frac{1}{a} + \frac{n(a - b)}{ab(n + 1)}\)

\(H_n = \frac{ab(n + 1)}{na + b}\)

36.3.3. Relation between A.M, G. M. and H. M. between two real and unequal quantities

Let \(a\) and \(b\) be two real, positive and unequal quantities and \(A, G\) and \(H\) be the single A. M., G. M. and H. M. respectively.

Then, \(A = \frac{a + b}{2}, G = \sqrt{ab}, H = \frac{2ab}{a + b}\)

Now, \(AH = \frac{a + b}{2}.\frac{2ab}{a + b} = ab = G^2 \therefore \frac{G}{A} = \frac{H}{G}\)

Hence, \(A, G\) and \(H\) are in G. P.

Again. \(A - G = \frac{a + b}{2} - \sqrt{ab} = \frac{a + b - 2\sqrt{ab}}{2}\)

\(= \frac{(\sqrt{a} - \sqrt{b})^2}{2} > 0~[\because a\ne 0]\)

Thus, \(A - G > 0\) or \(A > G\)

Since, \(\frac{H}{G} = \frac{G}{A}\) but \(\frac{G}{A} < 1 \therefore \frac{H}{G} < 1\)

Thus, \(A > G > H\)

For equal \(a\) and \(b\) it can be easily verified that \(A = G = H\)

36.4. Problems

  1. If \(n\) arithmetic means are inserted between \(20\) and \(80\) such that first mean : last mean = 1 : 3. Find \(n\).
  2. Prove that the sum of \(n\) arithmetic means between two given numbers is \(n\) times the single arithmetic mean between them.
  3. Between two numbers whose sum is \(\frac{13}{6}\), 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 \(n, \frac{a^{n + 1} + b^{n + 1}}{a^n + b^n}, a\ne b\) is the A. M. of \(a\) and \(b\).
  5. Insert \(4\) G. M. between \(5\) and \(160\).
  6. Show that the product of \(n\) geometric means inserted between two positive quantities is equal to the \(n\) the power of the single geometric means inserted between them.
  7. Insert \(6\) harmonic means between \(3\) and \(\frac{6}{23}\).
  8. If the A. M. and G. M. between two numbers be \(5\) and \(3\) 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 \(2 + \sqrt{3}: 2 - \sqrt{3}\).
  10. If \(a\) be one A. M. and \(g_1\) abd \(g_2\) be two G. M. between \(b\) and \(c\), prove that \(g_1^{3} + g_2^3 = 2abc\)
  11. If \(a, b, c\) be in G. P. and \(x, y\) be the A. M. between \(a, b\) and \(b, c\) respectively, show that \(\frac{a}{x} + \frac{b}{y} = 2, \frac{1}{x} + \frac{1}{y} = \frac{2}{b}\)
  12. If \(A\) be the A. M. and \(H\) be the H. M. between two quantities \(a\) and \(b\), prove that \(\frac{a - A}{a - H}.\frac{b - A}{b - H} = \frac{A}{H}\)
  13. If \(A_1, A_2\) be the A. M., \(G_1, G_2\) be the G. M. and \(H_1, H_2\) be the H. M. between any two quantities, show that \(\frac{G_1G_2}{H_1H_2} = \frac{A_1 + A_2}{H_1 + H_2}\)
  14. The arithmetic mean of two numbers exceed their geometric mean by \(\frac{3}{2}\) and the geometric mean exceeds their harmonic mean by \(\frac{6}{5}\), find the numbers.
  15. If \(a, b, c, d\) be four distinct quantities in H. P., show that (i) \(a + d > b + c\) (ii) \(ad > bc\)
  16. If three positive uneuqal quantities \(a, b, c\) be in H. P. prove that \(a^n + c^n > 2b^n\), where \(n\) is a positive integer.
  17. If \(x + y + z = 15\) if \(a, x, y, z, b\) are in A. P. and \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{5}{3}\) if \(a, x, y, z, b\) are in H. P. find \(a\) and \(b\).
  18. If \(x > 0\), prove that \(x + \frac{1}{x} \geq 2\)
  19. Insert \(8\) A. M. between \(5\) and \(32\).
  20. Insert \(7\) A. M. between \(2\) and \(34\).
  21. Insert \(17\) A. M. between \(\frac{7}{2}\) and \(-\frac{83}{2}\).
  22. Between \(1\) and \(31, n\) A. M. are inserted such that ratio of the \(7\) th and \((n - 1)\) th means is \(5:9\), find \(n\).
  23. Find the relation between \(x\) and \(y\) in order that \(r\) th mean between \(x\) and \(2y\) may be the same as \(r\) th mean between \(2x\) and \(y\); if \(n\) arithmetic means are inserted in each case.
  24. Insert \(7\) geometric means between \(2\) and \(162\).
  25. Insert \(6\) geometric means between \(\frac{8}{27}\) and \(\frac{-81}{16}\).
  26. If odd numbers of geometric means are inserted between two given quantities \(a\) and \(b\), show that the middle geometric mean is \(\sqrt{ab}\).
  27. Insert four harmonic means between \(1\) and \(\frac{1}{11}\).
  28. \(n\) harmonic means have been inserted between \(1\) and \(4\) such that first mean:last mean = \(1:3\), find \(n\).
  29. Find \(n\) such \(\frac{a^{n+1}+b^{n+1}}{a^n + b^n}\) may be a single harmonic mean between \(a\) and \(b\).
  30. If \(H_1, H_2, ..., H_n\) be \(n\) harmonic means between \(a\) and \(b\), prove that \(\frac{H_1 + a}{H_1 - a} + \frac{H_n + b}{H_n - b} = 2n\)
  31. If \(A\) be the A. M. and \(G\) be the G. M. between two numbers, show that the numbers are \(A + \sqrt{A^2 - G^2}\) and \(A - \sqrt{A^2 - G^2}\)
  32. If the ratio of A. M. and G. M. between two numbers \(a\) and \(b\) is \(m:n\), prove that \(a:b = m + \sqrt{m^2 - n^2}:m - \sqrt{m^2 - n^2}\)
  33. If one G. M. \(G\) and two A. M. \(p\) and \(q\) be inserted between two given quantities, prove that \(G^2 = (2p - q)(2q - p)\)
  34. If one A. M. \(A\) and two G. M. \(p\) and \(q\) be inserted between two numbers, show that \(\frac{p^2}{q} + \frac{q^2}{p} = 2A\)
  35. If A. M. between \(a\) and \(b\) is equal to \(m\) times the H. M., prove that \(a:b = \sqrt{m}+\sqrt{m - 1}:\sqrt{m}-\sqrt{m - 1}\)
  36. If \(9\) arithmetic means and \(9\) harmonic means be inserted between \(2\) and \(3\), prove that \(A + \frac{6}{H} = 5\), where \(A\) is any arithmetic mean and \(H\), the corresponding harmonic mean.
  37. If \(a\) is the A. M. between \(b\) and \(c\), \(b\) the G. M. between \(a\) and \(c\), then show that \(c\) is the H. M. between \(a\) and \(b\).
  38. If \(a_1, a_2\) be the two A. M., \(g_1, g_2\) be the two G. M. and \(h_1, h_2\) be the two H. M. between any two numbers \(x\) and \(y\), show that \(a_1h_2 = a_2h_1 = g_1g_2 = xy\)
  39. If between any two quantities, there be inserted \(2n - 1\) arithmetic, geometric and harmonic means, show that \(n\) th means inserted are in G. P.
  40. The A. M. between two numbers exceed their G. M. by \(2\) and the G. M. exceeds the H. M. by \(\frac{8}{5}\), find the numbers.
  41. The harmonic mean of two numbers is \(4\), their A. M. \(A\) and G. M. \(G\) satisfy the relation \(2A + G^2 =27\). Find the numbers.
  42. If \(a, b, c\) are in H. P., prove that i. \(a^2 + c^2 > 2b^2\) ii. \(a^5 + c^5 > 2b^5\)
  43. Prove that \(b^2 > = < ac\) according as \(a, b, c\) are in A. P., G. P. and H. P.