Question

How do you find the geometric mean between $5$ and $20$?

Answer 1

Hint: The geometric mean between two numbers is defined as the number which when written between the two numbers forms a sequence known as the geometric progression. We can assume the geometric mean to be x, such that the GP will become $5,x,20$. We know that the geometric progression is a sequence in which the ratio of two consecutive terms is always a constant. Therefore, in the GP $5,x,20$ the condition will become \[\dfrac{x}{5}=\dfrac{20}{x}\]. On solving this equation, we will get the value of x, and hence the required value of the geometric mean.

Complete step by step solution:
Let the geometric mean between the given numbers be x.
We know that the geometric mean between two numbers is a number which when written between the two numbers, forms a sequence known as the geometric progression, or the GP. Therefore, the GP in this case will become $5,x,20$. Now, since the ratio of two consecutive in a GP is a constant, for the GP $5,x,20$ we can write
$\Rightarrow \dfrac{x}{5}=\dfrac{20}{x}$
Multiplying both the sides by x, we get
$\Rightarrow \dfrac{{{x}^{2}}}{5}=20$
Now, multiplying both the sides by $5$, we get
$\Rightarrow {{x}^{2}}=100$
On solving the above equation, we get
$\Rightarrow x=\pm 10$

Hence, for the numbers $5$ and $20$, we obtained two geometric means as $10$ and $-10$.

Note:It is a common misconception that the geometric mean between two numbers a and b is given by $\sqrt{ab}$. This is not true since we miss the negative value of the geometric mean between the numbers. So we can remember the geometric mean as $\pm \sqrt{ab}$.