Question

If G.M $= 18$ and A.M $= 27$ , then H.M is
$A)\dfrac{1}{{18}}$
$B)\dfrac{1}{{12}}$
$C)12$
$D)9\sqrt 6$

Answer 1

First, we need to know about the concepts of arithmetic mean, geometric mean, and harmonic mean.
After known, the concepts derive the formulas for these three mean values,
Finally compare the relationship between these AM, GM, HM, and generalize the common formula.
Thus, substitute the given values in the formula and we get the required HM.

Complete step by step answer:
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 given total number of terms.
Thus, we get $AM = \dfrac{{a + b}}{2}$ where a and b are the sum of the terms and two is the total count.
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.
Thus, we get $GM = \sqrt {ab}$
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.
Thus, we get $HM = \dfrac{2}{{\dfrac{1}{a} + \dfrac{1}{b}}} \Rightarrow \dfrac{{2ab}}{{a + b}}$
Finally, substitute $AM = \dfrac{{a + b}}{2}$ and $ab = G{M^2}$ IN HM, then we get $HM = \dfrac{{2ab}}{{a + b}} \Rightarrow \dfrac{{G{M^2}}}{{AM}} \Rightarrow G{M^2} = AM \times HM$
Substituting the given values G.M $= 18$ and A.M $= 27$ in the above equation we get, $G{M^2} = AM \times HM \Rightarrow {18^2} = 27 \times HM$
Further solving the values, we get $HM = \dfrac{{{{18}^2}}}{{27}} \Rightarrow \dfrac{{324}}{{27}} = 12$

So, the correct answer is “Option C”.

Note: An arithmetic progression can be given by $a,(a + d),(a + 2d),(a + 3d),...$ where $a$ is the first term and $d$ is a common difference.
A geometric progression can be given by $a,ar,a{r^2},....$ where $a$ is the first term and $r$ is a common ratio.
Harmonic progress is the reciprocal of the given arithmetic progression which is the form of $HP = \dfrac{1}{{[a + (n - 1)d]}}$ where $a$ is the first term and $d$ is a common difference and n is the number of AP.