Question

The product of three geometric means between 4 and \[\dfrac{1}{4}\] will be
A. 4
B. 2
C. -1
D. 1

Answer 1

Hint: The geometric mean is calculated by raising the product of a series of numbers to the inverse of the total length of the series. As per given statement, there are three geometric means and to find the product of the geometric means we must form a geometric series according to the given data i.e., of the form \[a + ar + a{r^2} + a{r^3} + a{r^4} + ...\] in which we can easily find the product of 4 and \[\dfrac{1}{4}\] by applying the formula of geometric sequence.
Formula used:
Geometric series sequence form: \[a + ar + a{r^2} + a{r^3} + a{r^4} + ...\]
a is the first term.
r is the common ratio.
 \[{a_n} = a{r^{n - 1}}\]
 \[{a_n}\] is the nth term of the sequence
a is the first term of the sequence.
r is the common ratio of the sequence.

Complete step by step solution:
As per given data, there are three geometric means, let \[{G_1},{G_2},{G_3}\] be three geometric means.
Let, \[4,{G_1},{G_2},{G_3},\dfrac{1}{4}\] be the geometric series according to the given data.
Hence, here \[a = 4\]
 \[a{r^4} = \dfrac{1}{4}\]
 \[ \Rightarrow \] \[4\left( {{r^4}} \right) = \dfrac{1}{4}\]
 \[ \Rightarrow \] \[{r^4} = \dfrac{1}{{16}}\]
 \[ \Rightarrow \] \[r = \dfrac{1}{2}\]
According to the given data, we need to find the product of three geometric means between 4 and \[\dfrac{1}{4}\] , hence
Product of \[{G_1},{G_2},{G_3} = ar\left( {a{r^2}} \right)\left( {a{r^3}} \right)\]
= \[{a^3}{r^6}\]
= \[{\left( {a{r^2}} \right)^3}\]
Now, substitute the values of a and r as:
= \[{\left( {4 \times {{\left( {\dfrac{1}{2}} \right)}^2}} \right)^3}\]
= \[{\left( {4 \times \dfrac{1}{4}} \right)^3}\]
= \[1\]
 \[ \Rightarrow \] \[{G_1},{G_2},{G_3} = 1\]
Therefore, the product of three geometric means between 4 and \[\dfrac{1}{4}\] is 1.
Hence, option D is the right answer.
So, the correct answer is “Option D”.

Note: The geometric mean is most useful when numbers in the series are not independent of each other. Hence, in the given question to find the product of three geometric means we must note that the geometric mean is given not geometric series values, hence while calculating we must note the formula to find the product i.e., \[{G_1},{G_2},{G_3} = ar\left( {a{r^2}} \right)\left( {a{r^3}} \right)\] .