Question

If \[{H_1}\] and \[{H_2}\] are two harmonic means between two positive numbers \[a\] and \[b\],\[\left( {a \ne b} \right)\]. \[A\] and \[G\] are the arithmetic and geometric means between \[a\] and \[b\]. Then what is the value of \[\dfrac{{{H_2} + {H_1}}}{{{H_1}{H_2}}}\]?
A. \[\dfrac{A}{G}\]
B. \[\dfrac{{2A}}{G}\]
C. \[\dfrac{A}{{{G^2}}}\]
D. \[\dfrac{{2A}}{{{G^2}}}\]

Answer 1

Hint: First, use the definition of harmonic progression and convert the sequence into the arithmetic progression. Then calculate the arithmetic mean and the geometric mean of the positive numbers \[a\] and \[b\]. After that, find the arithmetic means of the first 3 terms of the arithmetic progression and the last 3 terms of the arithmetic progression. Then add both arithmetic means and simplify the equation to reach the required answer.

Formula used:
If \[a,b\] and \[c\] are in the arithmetic progression, then \[2b = a + c\]
If \[a,b\] and \[c\] are in the geometric progression, then \[{b^2} = ac\]
If \[a,b\] and \[c\] are in the arithmetic progression, then the harmonic progression is \[\dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c}\]

Complete step by step solution:
Given:
\[{H_1}\] and \[{H_2}\] are two harmonic means between two positive numbers \[a\] and \[b\].
\[A\] and \[G\] are the arithmetic and geometric means between \[a\] and \[b\].
Let’s consider \[a,{H_1},{H_2},b\] are in harmonic progression.
Apply the definition of the arithmetic progression.
\[\dfrac{1}{a},\dfrac{1}{{{H_1}}},\dfrac{1}{{{H_2}}},\dfrac{1}{b}\] are in the arithmetic progression.
Since \[A\] and \[G\] are the arithmetic and geometric means between \[a\] and \[b\].
Then,
\[2A = a + b\] and \[G = \sqrt {ab} \]
\[ \Rightarrow \]\[2A = a + b\] and \[{G^2} = ab\]
From the above arithmetic progression, we get
\[\dfrac{1}{a},\dfrac{1}{{{H_1}}},\dfrac{1}{{{H_2}}}\] are in the arithmetic progression.
Then,
\[2\left( {\dfrac{1}{{{H_1}}}} \right) = \dfrac{1}{a} + \dfrac{1}{{{H_2}}}\] \[.....\left( 1 \right)\]
Also, \[\dfrac{1}{{{H_1}}},\dfrac{1}{{{H_2}}},\dfrac{1}{b}\] are in the arithmetic progression.
Then,
\[2\left( {\dfrac{1}{{{H_2}}}} \right) = \dfrac{1}{b} + \dfrac{1}{{{H_1}}}\] \[.....\left( 2 \right)\]
Now add the equations \[\left( 1 \right)\] and \[\left( 2 \right)\].
\[2\left( {\dfrac{1}{{{H_1}}} + \dfrac{1}{{{H_2}}}} \right) = \dfrac{1}{a} + \dfrac{1}{{{H_2}}} + \dfrac{1}{b} + \dfrac{1}{{{H_1}}}\]
\[ \Rightarrow \]\[2\left( {\dfrac{1}{{{H_1}}} + \dfrac{1}{{{H_2}}}} \right) - \left( {\dfrac{1}{{{H_1}}} + \dfrac{1}{{{H_2}}}} \right) = \dfrac{1}{a} + \dfrac{1}{b}\]
\[ \Rightarrow \]\[\dfrac{1}{{{H_1}}} + \dfrac{1}{{{H_2}}} = \dfrac{1}{a} + \dfrac{1}{b}\]
Simplify the above equation.
\[\dfrac{{{H_2} + {H_1}}}{{{H_1}{H_2}}} = \dfrac{{a + b}}{{ab}}\]
Substitute the values of \[a + b\] and \[ab\] in the above equation.
\[\dfrac{{{H_2} + {H_1}}}{{{H_1}{H_2}}} = \dfrac{{2A}}{{{G^2}}}\]
Hence the correct option is D.

Note: Students often get confused with the concept of harmonic progression.
A harmonic progression is a sequence of real numbers that are determined by taking the reciprocals of the terms in the arithmetic progression.