Question

Find the geometric mean of 4 and 25.

Answer 1

Hint: Geometric mean is mean or average. For taking the geometric mean of two numbers, take the square root of the products of two numbers.
Complete step-by-step answer:
The geometric mean is a mean or average which indicates the central tendency or typical value of a set of numbers by using the product of their values. The geometric mean is defined as the \[{{n}^{th}}\] root of the product of n numbers i.e. for a set of numbers \[{{x}_{1}},{{x}_{2}},{{x}_{3}}......{{x}_{n}}\], the geometric mean is defined as,
\[{{\left( \prod\limits_{i=1}^{n}{{{x}_{i}}} \right)}^{\dfrac{1}{n}}}=\sqrt[n]{{{x}_{1}}{{x}_{2}}......{{x}_{n}}}\]
For instance, if we are taking the geometric mean of two numbers then we need to take square root of their product. Similarly, if we need to find the geometric of three numbers then take cube root of their product and so on.
Now, we have been given two numbers 4 and 25. Let us take,
a = 4 and b = 25.
\[\therefore \] Geometric mean (GM) \[=\sqrt{ab}\]
                                               \[=\sqrt{4\times 25}=\sqrt{100}=10\]
Thus Geometric mean of 4 and 25 is 10.

Note: We can also define geometric mean as when a positive value is repeated in either the means or extreme position of a proportion, that value is redefined to as Geometric mean (or mean proportional) between the other two values.
\[\therefore \] Let x be geometric mean then,
\[\dfrac{4}{x}=\dfrac{x}{25}\] (Cross product property)
\[\begin{align}
  & {{x}^{2}}=100 \\
 & \therefore x=\sqrt{100}=10 \\
\end{align}\]
i.e. GM = 10.