Question

If \[{A_1},{A_2};{G_1},{G_2}\]and \[{H_1},{H_2}\]be two A.M.s, G.M.s and H.M.s between two numbers respectively, then \[\frac{{{G_1}{G_2}}}{{{H_1}{H_2}}} \times \frac{{{H_1} + {H_2}}}{{{A_1} + {A_2}}} = \]
A. \[1\]
B. \[0\]
C. \[2\]
D. \[3\]

Answer 1

Hint
The total number of data values is n, so multiply all the numbers together and calculate the nth root of the multiplied numbers. The arithmetic mean (AM) and the harmonic mean (HM) are multiplied to create the geometric mean (GM). The product of the two integers' roots provides the geometric mean. We always get to the conclusion that the two numbers in the given series are equal to one another when we take into account all the alternatives.
The arithmetic means of a list of non-negative real numbers are greater than or equal to the geometric means of the same list, according to the AM-GM inequality, also known as the inequality of arithmetic and geometric means. Only a possibility exists if each number on the list is the same for the two numbers.
Formula use:
To find the ratio of two numbers,
H.M of a and b \[ = \frac{{2ab}}{{(a + b)}}\]
G.M of a and b \[ = \sqrt {(ab)} \]
 \[AM \times HM{\rm{ }} = {\rm{ }}G{M^2}\]
The sum of n A.M.s is equal to \[n \times \]single A.M.
the product of n G.M.s is equal to single G.M. \[ \times n\]
Complete step-by-step solution
Assume a and b be two numbers.
The sum of n A.M.s is equal to \[n \times \]single A.M.
\[ = > {A_1} + {A_2} = 2 \times (\frac{{a + b}}{2})\]
This equation thus becomes
\[a + b\]
Similarly, the product of n G.M.s is equal to single G.M. \[ \times n\]
The equation becomes,
\[{G_1} \times {G_2} = {(\sqrt {ab} )^2}\]\[ = ab\]
The series of numbers in A.P is
\[\frac{1}{a},\frac{1}{{{H_1}}},\frac{1}{{{H_2}}},\frac{1}{b}\]
\[ = > \frac{1}{{{H_1}}} + \frac{1}{{{H_2}}} = \frac{1}{a} + \frac{1}{b}\]
This then becomes,
\[\frac{{a + b}}{{ab}}\]
\[ = > \frac{{{H_1}{H_2}}}{{{H_1} + {H_2}}} = \frac{{{G_1}{G_2}}}{{{A_1} + {A_2}}}\]
This equation is finally calculated as
\[ = > \frac{{{G_1}{G_2}}}{{{H_1}{H_2}}} = \frac{{{H_1} + {H_2}}}{{{A_1} + {A_2}}} = 1\]
Therefore, the correct option is A.
Note
The arithmetic mean is calculated by adding a set of numbers, dividing by their count, and then taking the result. The Geometric Mean is a unique kind of average in which the numbers are multiplied together and then their square, cube, or other root is taken (for two numbers, three numbers, etc.).
The geometric mean, also known as the geometric mean (GM), while the arithmetic mean is the average of two numbers.