Question

The geometric mean and harmonic mean of two non-negative observations are $10$and $8$respectively. Then what is the arithmetic mean of the observations equal to?
$A)4$
$B)9$
$C)12.5$
$D)25$

Answer 1

First, we need to know about the geometric mean, arithmetic mean, and harmonic mean.
Arithmetic mean is the average or mean of the given set of numbers which is computed by adding all the terms in the set of numbers and dividing the sum by the total number of terms in the given.
The geometric mean is the mean value or the central term in the set of numbers in the geometric progression. Geometric means of sequence with the n terms is computed as the nth root of the product of all the terms in the sequence taken.
The harmonic mean is one of the types of determining the average. It is computed by dividing the number of values in the sequence by the sum of reciprocals of the terms in the sequence.

Complete step by step answer:
Since as per the question, they were asking us to find the relation of the AM, GM, HM, and then we can conclude the required AM.
Relation between AM, GM, and HM can be expressed as $G.{M^2} = A.M \times H.M$
From the given that we have geometric observation is $10$and the harmonic observation is $8$
Substituting these values in the relation to get the required AM value,
Thus, we get, $G.{M^2} = A.M \times H.M \Rightarrow {10^2} = A.M \times 8$
With the help of division operation, we get, ${10^2} = A.M \times 8 \Rightarrow A.M = \dfrac{{100}}{8}$
Further solving, we get, $A.M = \dfrac{{100}}{8} \Rightarrow A.M = 12.5$
Hence option $C)12.5$ is correct.
There is no possibility of getting the other options, because if we get that then the given GM and HM values will get affected. So, options like A, B, and D are incorrect in the relations.

So, the correct answer is “Option C”.

Note: Now consider the two numbers a and b, where these numbers are greater than zero.
And the number of n terms in the sequence expressed as $\dfrac{{a + b}}{2}$is the expression of AM.
$\sqrt {ab}$is the expression of GM and $\dfrac{{2ab}}{{a + b}}$is the expression of HM. By applying that AM as $\dfrac{{a + b}}{2}$and $ab$as the $G{M^2}$. Substitute in the HM we get, $\dfrac{{2ab}}{{a + b}} \Rightarrow \dfrac{{G{M^2}}}{{AM}}$
Hence the relation $G.{M^2} = A.M \times H.M$