Question

Calculate the geometric mean of 3 and 27.
(A) 3
(B) 6
(C) 9
(D) 12

Answer 1

Here, you are asked to find the geometric mean of the given two numbers that are 3 and 7, so in order to find it, first you need to understand the meaning of geometric mean. First, consider a set of numbers and find the geometric mean of those numbers so you have what you need in a general form, then reduce your set to two numbers and substitute the elements of the set by the numbers given to you.

Complete step by step answer:
Geometric mean is another method of calculating the average of numbers. Usually while finding the average of numbers using the arithmetic mean, you add all the numbers and then divide the obtained sum by the total number of numbers that you have. In geometric mean, what you do is, you find the product of the numbers that you have and then take the nth root of the obtained product, where n is the number of numbers that you used.

Let us consider a set of numbers as (a0, a1, a2 … an-1). As per the definition provided above, mathematically, we can write geometric mean as $GM = \sqrt[n]{{{a_0}{a_1}{a_2}...{a_{n - 1}}}}$.So, our set will be (3, 27), here ${a_0} = 3$, ${a_1} = 27$ and $n = 2$. Therefore, geometric mean of 3 and 27 will be,
$GM = \sqrt {3 \times 27} \\\ \Rightarrow GM = {\left( {3 \times 27} \right)^{\dfrac{1}{2}}} \\\ \Rightarrow GM = {\left( {3 \times {3^3}} \right)^{\dfrac{1}{2}}} \\\ \Rightarrow GM = {\left( {{3^4}} \right)^{\dfrac{1}{2}}} \\\ \therefore GM = 9 \\\$
Therefore, the geometric mean of 3 and 27 is 9 and thus option C is correct.

Note: We have discussed in brief about the method of calculating arithmetic mean and geometric mean, so you are supposed to remember both the methods. Note that you can find the geometric mean by taking the sum of logarithms of each number given to you, dividing the obtained sum by the number of numbers you are given and finally taking the antilog of the obtained number.