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**Hint**: Let us assume two numbers, a and b. We know that AM is arithmetic mean and is given as 9 in the question, therefore we get $ \dfrac{a+b}{2}=9 $ . Also, we know that HM is a harmonic mean and we are given that HM is 4, so we get $ \dfrac{2ab}{a+b}=4 $ .

Now, we know that GM is the geometric mean and it is given as $ GM=\sqrt{ab} $ . So, to find GM we need to find a value of $ ab $ and then substitute in $ GM=\sqrt{ab} $ to get the value of GM. For getting a value of $ ab $ , evaluate the equations of AM and GM.

**:**

__Complete step-by-step answer__Consider two numbers ‘a’ and ‘b’.

The arithmetic mean of a & b is defined as:

$ AM=\dfrac{a+b}{2} $

Since AM of a & b is given as 9.

Therefore, we can write:

$ \dfrac{a+b}{2}=9 $

$ a+b=18......(1) $

Now, the Harmonic mean of a & b is defined as:

$ HM=\dfrac{2ab}{a+b} $

Since HM of a & b is given as 4.

Therefore, we can write:

$ \dfrac{2ab}{a+b}=4 $

$ \begin{align}

& \Rightarrow ab=\dfrac{4\left( a+b \right)}{2} \\

& \Rightarrow ab=\dfrac{4\times 18}{2} \\

& \Rightarrow ab=36......(2) \\

\end{align} $

To calculate the geometric mean of a & b that is defined as:

$ GM=\sqrt{ab}......(3) $

Substitute the value of $ ab $ from equation (2) in equation (3), we get:

$ \begin{align}

& GM=\sqrt{36} \\

& GM=6 \\

\end{align} $

**Note**: We can also find geometric mean of two number by another method stated below:

The geometric mean of any two positive numbers can be given as $ {{\left( GM \right)}^{2}}=\left( AM \right)\left( HM \right) $

For the above question:

$ \begin{align}

& {{\left( GM \right)}^{2}}=\left( 9 \right)\left( 4 \right) \\

& =36

\end{align} $

$ \Rightarrow GM=6 $ $ \Rightarrow GM=6 $

Also, to check whether the answer is correct, always remember GM lies between harmonic mean (HM) and arithmetic mean (AM) of two numbers.

i.e. $ HM\le GM\le AM $

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