Question

The GM of numbers 4, 5, 10, 20, 25 is
(a) 12.8
(b) 10
(c) 7.8
(d) none of these

Answer 1

Hint: Here, we will use the formula for the geometric mean of a certain number of numbers. The geometric mean of ‘n’ numbers is given as:
${{\left( \prod\limits_{i=1}^{n}{{{x}_{i}}} \right)}^{\dfrac{1}{n}}}={{\left( {{x}_{1}}.{{x}_{2}}.....{{x}_{n}} \right)}^{\dfrac{1}{n}}}$

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 nth 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:
 ${{\left( \prod\limits_{i=1}^{n}{{{x}_{i}}} \right)}^{\dfrac{1}{n}}}={{\left( {{x}_{1}}.{{x}_{2}}.....{{x}_{n}} \right)}^{\dfrac{1}{n}}}...........\left( 1 \right)$
The $\prod\limits_{{}}^{{}}{{}}$symbol in the formula is the mathematical notation for product in the same way that the notation for summation is $\sum\limits_{{}}^{{}}{{}}$.
The central number in a geometric progression is also the geometric mean of a sequence of numbers.
Here, we have been given five numbers. The numbers are: 4, 5, 10, 20 and 25.
So, here n = 5.
${{x}_{1}}=4,{{x}_{2}}=5,{{x}_{3}}=10,{{x}_{4}}=20\text{ and }{{x}_{5}}=25$
On substituting these values in equation (1), we get:
$\begin{align}
  & {{\left( \prod\limits_{i=1}^{5}{{{x}_{i}}} \right)}^{\dfrac{1}{5}}}={{\left( 4\times 5\times 10\times 20\times 25 \right)}^{\dfrac{1}{2}}} \\
 & \Rightarrow {{\left( \prod\limits_{i=1}^{5}{{{x}_{i}}} \right)}^{\dfrac{1}{5}}}={{\left( 100000 \right)}^{\dfrac{1}{5}}} \\
 & {{\left( \prod\limits_{i=1}^{5}{{{x}_{i}}} \right)}^{\dfrac{1}{5}}}={{\left( {{10}^{5}} \right)}^{\dfrac{1}{5}}}=10 \\
\end{align}$
Therefore, the geometric mean of the given numbers is 10.
Hence, option (b) is the correct answer.

Note: Students should note here that the middle term of the given set of numbers is also 10. It means that the middle term in any set of numbers is their geometric mean. Students should remember the formula used to find the geometric mean to avoid unnecessary mistakes.