# How do you find the geometric means in each sequence 1/24, __, __, __, 54?

Jan 1, 2017

The three geometric means are:

$\frac{1}{4}$, $\frac{3}{2}$ and $9$

#### Explanation:

The general term of a geometric sequence can be written as:

${a}_{n} = a \cdot {r}^{n - 1}$

where $a$ is the initial term and $r$ is the common ratio.

If ${a}_{1} = \frac{1}{24}$ and ${a}_{5} = 54$ then we find:

${r}^{4} = \frac{a {r}^{4}}{a {r}^{0}} = {a}_{5} / {a}_{1} = \frac{54}{\frac{1}{24}} = 54 \cdot 24 = 1296 = {6}^{4}$

This has two Real solutions and two non-Real Complex solutions:

$r = \pm 6 \text{ }$ or $\text{ } r = \pm 6 i$

Since the question asks about geometric means and the given terms are positive, we can probably assume that we want the positive common ratio $r = 6$.

Hence the sequence is:

$\frac{1}{24} , \textcolor{b l u e}{\frac{1}{4}} , \textcolor{b l u e}{\frac{3}{2}} , \textcolor{b l u e}{9} , 54$