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Hint: We calculate the GM or the geometric mean between any two numbers a and b by first taking the product of the given numbers a and b, and after we have calculated the product, we will be taking the square root of it, to obtain the required GM or the geometric mean.

Let us begin by understanding the mathematical meaning of geometric mean. A geometric means is mathematically denoted as the nth root product of the given ‘n’ numbers.

Let us consider the given two numbers a and b. In order to find their geometric mean, we will first find their product and denote it by P.

$

\Rightarrow P = {\text{Product of given two numbers}} \\

\Rightarrow P = a \times b \\

\Rightarrow P = ab \\

$

Now, we will take the square root of the obtained product to find the value of the geometric mean.

$

GM = \sqrt P \\

= \sqrt {ab} \\

$

Now, let us try to generalize this formula for geometric mean for ‘n’ numbers. For that, let us first consider that there are three numbers, a, b and c and you need to find the geometric mean between them.

We look at the mathematical form of the meaning of the geometric mean stated above.

Using that definition we get the geometric mean of 3 given numbers is the “cube root of the product of 3 number”.

Thus,

If there are ‘n’ numbers given to you, all you need to do is take their product and then apply nth root to the result obtained.

Let us take an example to better understand this formula obtained.

Let us consider the numbers 1,2, 3, 4 and 5.

There are 5 numbers in total.

Then to calculate the GM or the geometric mean, first multiply the five numbers.

$

\Rightarrow P = {\text{Product of 1, 2, 3, 4 and 5}} \\

\Rightarrow P = 1 \times 2 \times 3 \times 4 \times 5 \\

\Rightarrow P = 120 \\

$

Now, we will take the fifth root of the obtained product to find the value of the geometric mean between the given numbers.

\[

GM = \sqrt[5]{P} \\

= \sqrt[5]{{120}} \\

\]

Hence, we can say that the geometric mean between the two numbers a and b is $\sqrt {ab} $. Thus, option (A) is the correct option.

Note: Instead of mugging up the formulas, try to understand how that formula is obtained. Although this question is a known formula but if, in case, you forget the exact formula, you should be able to recall it, after understanding the reason behind getting that formula. Also note that as by the word mean you might think that you need to divide by 2, but do not forget that geometric mean and mean are two different things, and hence the formulas are different for both of them.

__Complete step-by-step solution__Let us begin by understanding the mathematical meaning of geometric mean. A geometric means is mathematically denoted as the nth root product of the given ‘n’ numbers.

Let us consider the given two numbers a and b. In order to find their geometric mean, we will first find their product and denote it by P.

$

\Rightarrow P = {\text{Product of given two numbers}} \\

\Rightarrow P = a \times b \\

\Rightarrow P = ab \\

$

Now, we will take the square root of the obtained product to find the value of the geometric mean.

$

GM = \sqrt P \\

= \sqrt {ab} \\

$

Now, let us try to generalize this formula for geometric mean for ‘n’ numbers. For that, let us first consider that there are three numbers, a, b and c and you need to find the geometric mean between them.

We look at the mathematical form of the meaning of the geometric mean stated above.

Using that definition we get the geometric mean of 3 given numbers is the “cube root of the product of 3 number”.

Thus,

If there are ‘n’ numbers given to you, all you need to do is take their product and then apply nth root to the result obtained.

Let us take an example to better understand this formula obtained.

Let us consider the numbers 1,2, 3, 4 and 5.

There are 5 numbers in total.

Then to calculate the GM or the geometric mean, first multiply the five numbers.

$

\Rightarrow P = {\text{Product of 1, 2, 3, 4 and 5}} \\

\Rightarrow P = 1 \times 2 \times 3 \times 4 \times 5 \\

\Rightarrow P = 120 \\

$

Now, we will take the fifth root of the obtained product to find the value of the geometric mean between the given numbers.

\[

GM = \sqrt[5]{P} \\

= \sqrt[5]{{120}} \\

\]

Hence, we can say that the geometric mean between the two numbers a and b is $\sqrt {ab} $. Thus, option (A) is the correct option.

Note: Instead of mugging up the formulas, try to understand how that formula is obtained. Although this question is a known formula but if, in case, you forget the exact formula, you should be able to recall it, after understanding the reason behind getting that formula. Also note that as by the word mean you might think that you need to divide by 2, but do not forget that geometric mean and mean are two different things, and hence the formulas are different for both of them.

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