Question

If two arithmetic means ${{A}_{1}},{{A}_{2}}$, two geometric means ${{G}_{1}},{{G}_{2}}$ and two harmonic means ${{H}_{1}},{{H}_{2}}$ are inserted between two numbers p and q then
A. $\dfrac{{{G}_{1}}{{G}_{2}}}{{{H}_{1}}{{H}_{2}}}=\dfrac{\left( {{A}_{1}}+{{A}_{2}} \right)}{\left( {{H}_{1}}+{{H}_{2}} \right)}$
B. $\dfrac{\left( {{G}_{1}}+{{G}_{2}} \right)}{\left( {{H}_{1}}+{{H}_{2}} \right)}=\dfrac{\left( {{A}_{1}}{{A}_{2}} \right)}{\left( {{H}_{1}}{{H}_{2}} \right)}$
C. $\dfrac{{{G}_{1}}{{G}_{2}}}{{{H}_{1}}{{H}_{2}}}=\dfrac{\left( {{A}_{1}}-{{A}_{2}} \right)}{\left( {{H}_{1}}-{{H}_{2}} \right)}$
D. $\dfrac{\left( {{A}_{1}}+{{A}_{2}} \right)}{\left( {{H}_{1}}+{{H}_{2}} \right)}={{G}_{1}}{{G}_{2}}{{H}_{1}}{{H}_{2}}$

Answer 1

Hint: We first use the relation between the numbers and their arithmetic means, geometric means and harmonic means. We find the equations and simplify to place the expressions and find the solution between the means.

Complete step-by-step answer:
Two numbers p and q, we have two arithmetic means ${{A}_{1}},{{A}_{2}}$, two geometric means ${{G}_{1}},{{G}_{2}}$ and two harmonic means ${{H}_{1}},{{H}_{2}}$.
So, arithmetic means ${{A}_{1}},{{A}_{2}}$ in between p and q. The condition gives us ${{A}_{1}}+{{A}_{2}}=p+q$.
Similarly, geometric means ${{G}_{1}},{{G}_{2}}$ in between p and q. The condition gives us ${{G}_{1}}{{G}_{2}}=pq$.
And harmonic means ${{H}_{1}},{{H}_{2}}$ in between p and q.
We know that if certain numbers are in harmonic series, then the reciprocal numbers are in arithmetic series. Therefore, arithmetic means $\dfrac{1}{{{H}_{1}}},\dfrac{1}{{{H}_{2}}}$ in between $\dfrac{1}{p}$ and $\dfrac{1}{q}$.
The condition gives us $\dfrac{1}{{{H}_{1}}}+\dfrac{1}{{{H}_{2}}}=\dfrac{1}{p}+\dfrac{1}{q}$.
We have three conditions and we use them to find the relation.
We simplify $\dfrac{1}{{{H}_{1}}}+\dfrac{1}{{{H}_{2}}}=\dfrac{1}{p}+\dfrac{1}{q}$.
$\begin{align}
  & \dfrac{1}{{{H}_{1}}}+\dfrac{1}{{{H}_{2}}}=\dfrac{1}{p}+\dfrac{1}{q} \\
 & \Rightarrow \dfrac{{{H}_{1}}+{{H}_{2}}}{{{H}_{1}}{{H}_{2}}}=\dfrac{p+q}{pq} \\
\end{align}$
Now we put the values ${{A}_{1}}+{{A}_{2}}=p+q$ and ${{G}_{1}}{{G}_{2}}=pq$ to get
$\begin{align}
  & \dfrac{{{H}_{1}}+{{H}_{2}}}{{{H}_{1}}{{H}_{2}}}=\dfrac{p+q}{pq} \\
 & \Rightarrow \dfrac{{{H}_{1}}+{{H}_{2}}}{{{H}_{1}}{{H}_{2}}}=\dfrac{{{A}_{1}}+{{A}_{2}}}{{{G}_{1}}{{G}_{2}}} \\
 & \Rightarrow \dfrac{{{G}_{1}}{{G}_{2}}}{{{H}_{1}}{{H}_{2}}}=\dfrac{{{A}_{1}}+{{A}_{2}}}{{{H}_{1}}+{{H}_{2}}} \\
\end{align}$
Therefore, the correct option is A.
So, the correct answer is “Option A”.

Note: We must remove the numbers p and q from the expression of means. The means are also related individually for inequality form where $A.M.\ge G.M.\ge H.M.$ in case of inequality relation we can use that.