Question

If the geometric mean of $x,16,50$ is 20, then the value of $x$ is

1. 40
2. 20
3. 10

Answer 1

We know that geometric mean of ${a_1},{a_2}.....{a_n}$ is given as ${\left( {{a_1}.{a_2}.....{a_n}} \right)^{\dfrac{1}{n}}}$. We will substitute the values of ${a_1},{a_2},{a_3}$ and $n = 3$ in the given expression. Then equate it to 20. Take the cube on both sides and solve the equation for the value of $x$.

Complete step-by-step answer:
We are given that the geometric mean of $x,16,50$ is 20.
We know that the geometric mean of ${a_1},{a_2}.....{a_n}$ is given as ${\left( {{a_1}.{a_2}.....{a_n}} \right)^{\dfrac{1}{n}}}$
Then the geometric mean of $x,16,50$ is given as
${\left( {x.16.50} \right)^{\dfrac{1}{3}}}$
We are given that the geometric mean is equal to 20.
$\Rightarrow$ ${\left( {800x} \right)^{\dfrac{1}{3}}} = 20$
Take cube root on both sides.
$\Rightarrow$ $800x= {20^3}$
$\Rightarrow$ $800x = 20 \times 20 \times 20$
Divide both sides by 800
$\Rightarrow$ x =$\dfrac{{20 \times 20 \times 20}}{{800}}$
$\Rightarrow$ x = $10$
The value of $x$ is 10.
Hence, option (3) is correct.

Note: Geometric mean is a type of an average where we multiply the numbers and then take the ${n^{th}}$ root of the numbers, where $n$ is the number of terms. Geometric means help us to compare things with different properties.