Question

The geometric mean of the observations 2, 4, 8, 16, 32, 64 is
$
  {\text{A}}{\text{. }}{{\text{2}}^{\dfrac{5}{2}}} \\
  {\text{B}}{\text{. }}{{\text{2}}^{\dfrac{7}{2}}} \\
  {\text{C}}{\text{. 33}} \\
  {\text{D}}{\text{. None of these}} \\
$

Answer 1

Hint: - Geometric Mean (or GM) : It is a type of mean that indicates the central tendency of a set of numbers by using the product of their values. It is defined as the nth root of the product of n numbers. In this question we have to first observe the values that are used in the formula of Geometric Mean(GM) and then compute it.

Complete step-by-step answer:
 $ {\overline {\text{x}} _{geom}} = {\text{ }}_{}^n\sqrt {\prod\limits_{i = 1}^n {{x_i}} } \\
   \Rightarrow {\text{ }}{\overline {\text{x}} _{geom}}{\text{ }} = {\text{ }}_{}^n\sqrt {{x_1}.{x_2}.......{x_n}} \\
  {\text{ where,}} \\
  {\overline {\text{x}} _{geom}}{\text{ is the geometric mean(GM)}} \\
  n{\text{ is the total number of observations}} \\
  _{}^n\sqrt {\prod\limits_{i = 1}^n {{x_i}} } {\text{ is the }}{{\text{n}}^{th}}{\text{ square root of the product of the given numbers}} \\
$
Here observations are $2,4,8,16,32,64$
There are total 6 observations i.e., $n = 6$
$
  {\text{And the observations are :}} \\
  {{\text{x}}_1} = 2{\text{ }} \\
  {{\text{x}}_2} = 4{\text{ }} \\
  {{\text{x}}_3} = 8{\text{ }} \\
  {{\text{x}}_4} = 16{\text{ }} \\
  {{\text{x}}_5} = 32 \\
  {{\text{x}}_6} = 64{\text{ }} \\
 $

  $ \Rightarrow {\overline {\text{x}} _{geom}}{\text{ }} = {\text{ }}_{}^6\sqrt {\prod\limits_{i = 1}^6 {{x_i}} } \\
  {\text{On putting values of observations, we get}} \\
   \Rightarrow {\overline {\text{x}} _{geom}}{\text{ = }}_{}^6\sqrt {2.4.8.16.32.64} \\
  {\text{Now we can rewrite under root terms in terms of power of two in above equation }} \\
   \Rightarrow {\overline {\text{x}} _{geom}}{\text{ = }}_{}^6\sqrt {{2^1}{{.2}^2}{{.2}^3}{{.2}^4}{{.2}^5}{{.2}^6}} \\
   \Rightarrow {\overline {\text{x}} _{geom}}{\text{ = }}_{}^6\sqrt {{2^{(1 + 2 + 3 + 4 + 5 + 6)}}} {\text{ \{ }}\because {{\text{a}}^m}{\text{.}}{{\text{a}}^n}{\text{ = }}{{\text{a}}^{(m + n)}}\} \\
   \Rightarrow {\overline {\text{x}} _{geom}}{\text{ = }}{{\text{2}}^{\dfrac{{21}}{6}}} \\
   \Rightarrow {\overline {\text{x}} _{geom}}{\text{ = }}{{\text{2}}^{\dfrac{7}{2}}} \\
  {\text{Hence, option B}}{\text{. is correct}} \\
$

Note:- Whenever you get this type of question the key concept of solving is you have to know the Geometric Mean (GM) formula i.e.,${\overline {\text{x}} _{geom}} = _{}^n\sqrt {\prod\limits_{i = 1}^n {{x_i}} } $and interpretation of this formula means you have knowledge about how to interpret $n,{x_1},{x_2}.....{x_n}$. Put values of $n,{x_1},{x_2}.....{x_n}$ in GM formula and then solve it to the simplest form.