Question

What is the geometric mean of $5$ and $15$ ?

Answer 1

Hint: At first, we need to understand what Geometric means. The geometric mean of two numbers is the one which lies between the two numbers such that the three of them make a geometric sequence. We assume it to be “r”, and apply the common ratio formula, which will be $\dfrac{r}{5}=\dfrac{15}{r}$ . This gives $r=5\sqrt{3}$ .

Complete step-by-step answer:
A sequence is an enumerated collection of objects, especially numbers, in which repetitions are allowed and in which the order of objects matters. A sequence may be finite or infinite depending on the number of objects in the sequence. Sequences can be of various types such as arithmetic sequence, geometric sequence and so on. Sequences can be completely random as well. The ${{n}^{th}}$ term of a sequence is sometimes written as a function of n.
A geometric sequence is the one in which the ratio between the consecutive objects is a constant. It is called the common ratio of the geometric sequence. The geometric mean of two numbers is the one which lies between the two numbers such that the three of them make a geometric sequence. Let the geometric mean be r. By the rule of common ratios, we get,
$\begin{align}
  & \dfrac{r}{5}=\dfrac{15}{r} \\
 & \Rightarrow {{r}^{2}}=5\times 15 \\
 & \Rightarrow r=\sqrt{5\times 15}=5\sqrt{3} \\
\end{align}$
Thus, we can conclude that the geometric mean of the two given numbers is $5\sqrt{3}$ .

Note: We should not mistakenly apply the common difference formula instead of the common ratio formula, else we will get a totally different answer. We also have a shortcut to find the geometric mean, which is nothing but a predefined formula for it. $G.M=\sqrt{ab}$ where “a” and “b” are the two numbers and G.M is their geometric mean.