# 43. AM, GM, HM Relation and Problems

## 43.1. 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 $$11$$.

### 43.1.1. 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 $$A = \frac{a + b}{2}$$

### 43.1.2. 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. $$A_n = a + nd = \frac{a + nb}{n + 1}$$

## 43.2. 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 $$16$$.

### 43.2.1. 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 $$G^2 = ab \therefore G = \sqrt{ab} [\because G > 0]$$

### 43.2.2. 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. $$G_n = ar^n = a\left(\frac{b}{a}\right)^\frac{n}{n + 1}$$

## 43.3. 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.

### 43.3.1. 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 $$b = \frac{2ab}{a + b}$$

### 43.3.2. $$n$$ 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, $$\frac{1}{H_1} = \frac{1}{a} + d = \frac{bn + a}{ab(n + 1)}$$

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

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

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

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

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

### 43.3.3. 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 $$A = G = H$$

## 43.4. 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.