Question

If two arithmetic means ${\mathrm{A}}_1,{\mathrm{A}}_2$, two geometric means ${\mathrm{G}}_1,{\mathrm{G}}_2$ and two harmonic means ${\mathrm{H}}_1,{\mathrm{H}}_2$ are inserted between two numbers $\mathrm{p}$ and $\mathrm{q}$ then

• A. $\frac{G_1{G}_2}{H_1{H}_2}=\frac{\left(A_1+A_2\right)}{\left(H_1+H_2\right)}$
• B. $\frac{\left(G_1+G_2\right)}{\left(H_1+H_2\right)}=\frac{A_1{A}_2}{H_1{H}_2}$
• C. $\frac{G_1{G}_2}{H_1{H}_2}=\frac{\left(A_1-A_2\right)}{\left(H_1-H_2\right)}$
• D. $\frac{A_1+A_2}{H_1+H_2}=\frac{G_1{G}_2}{H_1{H}_2}$

Answer 1

The correct option is A

$\frac{G_1{G}_2}{H_1{H}_2}=\frac{\left(A_1+A_2\right)}{\left(H_1+H_2\right)}$

Explanation for the correct option:

Finding the relation:

Here, $\mathrm{p},{\mathrm{A}}_1,{\mathrm{A}}_2,\mathrm{q}$ are in A.P.,

$$\begin{gathered} \frac{{\mathrm{A}}_1+{\mathrm{A}}_2}{2}=\frac{\mathrm{p}+\mathrm{q}}{2} \\ \Rightarrow {\mathrm{A}}_1+{\mathrm{A}}_2=\mathrm{p}+\mathrm{q}...\left(\mathrm{i}\right) \end{gathered}$$

and $\mathrm{p},G_1,G_2,\mathrm{q}$ are in G.P.,

$$\begin{gathered} \sqrt{{\mathrm{G}}_1{\mathrm{G}}_2}=\sqrt{\mathrm{pq}} \\ \Rightarrow {\mathrm{G}}_1{\mathrm{G}}_2=\mathrm{pq}...\left(\mathrm{ii}\right) \end{gathered}$$

also, $\mathrm{p},{\mathrm{H}}_{1,}{\mathrm{H}}_2,\mathrm{q}$ are in H.P.,then

$\frac{1}{\mathrm{p}},\frac{1}{{\mathrm{H}}_1},\frac{1}{{\mathrm{H}}_2},\frac{1}{\mathrm{q}}$ are in A.P.

$$\begin{gathered} \begin{array}{rcl}& & \frac{{\displaystyle \frac{1}{{\mathrm{H}}_1}}+{\displaystyle \frac{1}{{\mathrm{H}}_2}}}{2}\end{array}\begin{array}{rcl}& =& \end{array}\begin{array}{rcl}& & \frac{{\displaystyle \frac{1}{\mathrm{q}}}+{\displaystyle \frac{1}{\mathrm{p}}}}{2}\end{array} \\ \Rightarrow \begin{array}{rcl}& & \frac{1}{{\mathrm{H}}_1}\end{array}\begin{array}{rcl}& & +\end{array}\begin{array}{rcl}& & \frac{1}{{\mathrm{H}}_2}\end{array}\begin{array}{rcl}& =& \end{array}\begin{array}{rcl}& & \frac{1}{\mathrm{q}}\end{array}\begin{array}{rcl}& & +\end{array}\begin{array}{rcl}& & \frac{1}{\mathrm{p}}\end{array} \end{gathered}$$

$\begin{array}{rcl}& \Rightarrow & \frac{{\mathrm{H}}_2+{\mathrm{H}}_1}{{\mathrm{H}}_1{\mathrm{H}}_2}=\frac{\mathrm{q}+\mathrm{p}}{\mathrm{pq}}...\left(\mathrm{iii}\right)\end{array}$

Putting $\mathrm{p}+\mathrm{q}$ and $\mathrm{pq}$ from equation $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$ in equation $\left(\mathrm{iii}\right)$, we get

$$\begin{gathered} \frac{{\mathrm{H}}_2+{\mathrm{H}}_1}{{\mathrm{H}}_1{\mathrm{H}}_2}=\frac{{\mathrm{A}}_1+{\mathrm{A}}_2}{{\mathrm{G}}_1{\mathrm{G}}_2} \\ \Rightarrow \frac{{\mathrm{G}}_1{\mathrm{G}}_2}{{\mathrm{H}}_1{\mathrm{H}}_2}=\frac{{\mathrm{A}}_1+{\mathrm{A}}_2}{{\mathrm{H}}_2+{\mathrm{H}}_1} \end{gathered}$$

Hence, the correct option is A.