Question

If the arithmetic, geometric and harmonic means between two positive real numbers be \[A,G\] and\[H\], then
A. \[{A^2} = GH\]
В. \[{H^2} = AG\]
C. \[G = AH\]
D. \[{G^2} = AH\]

Answer 1

Hint:
In this type of question, we must use formulas of various means. The three Pythagorean means are known as the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM). Also, if “a” and “b” are positive numbers, Arithmetic Mean \[AM = \frac{{(a + b)}}{2}\], Geometric Mean (GM) \[ = \sqrt {ab} \], and Harmonic Mean (HM) \[ = \frac{{(2ab)}}{{(a + b)}}\].
Formula used:
\[AM = \frac{{(a + b)}}{2}\]
(GM) \[ = \sqrt {ab} \]
(HM) \[ = \frac{{(2ab)}}{{(a + b)}}\]
Complete step-by-step solution:
Now we must determine the relationship between the arithmetic, geometric, and harmonic means of two distinct positive real numbers denoted by A, G, and H, respectively.
Assume “a” and “b” are two distinct positive real numbers, and we get.
Arithmetic Mean as,
\[ \Rightarrow AM = A = \frac{{(a + b)}}{2}\]--- (1)
Geometric Mean as,
\[ \Rightarrow GM = G = \sqrt {ab} \]--- (2)
Harmonic Mean as,
\[ \Rightarrow HM = H = \frac{{2ab}}{{(a + b)}}\]--- (3)
Now consider Arithmetic mean and Harmonic mean
\[{\rm{AH}} = \left( {\frac{{{\rm{a}} + {\rm{b}}}}{2}} \right)\left( {\frac{{2{\rm{ab}}}}{{{\rm{a}} + {\rm{b}}}}} \right)\]
Apply rule\[\left( {\frac{{2ab}}{{a + b}}} \right) = \frac{{2ab}}{{a + b}}\]:
\[ = \frac{{a + b}}{2} \cdot \frac{{2ab}}{{a + b}}\]
Cross cancel common factor \[\left( {a + b} \right)\]:
\[ = \frac{{2ab}}{2}\]
Cancel the common factor in the above equation:
\[{\rm{ = ab}}\]
Therefore, if the arithmetic, geometric and harmonic means between two positive real numbers be\[A,G\] and \[H\], then \[{\rm{AH}} = {\rm{ab}}\]
From equation (2),
To discover the relationship, consider the expression
\[{\rm{AH}} = {{\rm{G}}^2}\]
Since \[{\rm{AH}} = {{\rm{G}}^2}\], therefore, \[{\rm{A}},{\rm{G}}\] and \[{\rm{H}}\]are in G.P.
Hence, the correct answer is option D.
Note:
Students must remember the arithmetic mean, geometric mean, and harmonic mean formulas for this type of question. Students must exercise caution when simplifying the distinction between AM and GM, as well as GM and HM. Students must also remember that because both differences are in the form of squares, the differences must be greater than or equal to zero.