# How do you find the geometric means in the sequence 4, __, __, __, 324?

Oct 19, 2016

Middle terms are $\left\{12 , 36 , 108\right\}$ or $\left\{- 12 , 36 , - 108\right\}$

#### Explanation:

It appears that questioner wants to know three middle terms in the geometric sequence $\left\{4 , \ldots , \ldots , \ldots , 324\right\}$

Let the first term be $a$ and common ratio be $r$.

Then five terms are $\left\{a , a r , a {r}^{2} , a {r}^{3} , a {r}^{4}\right\}$

Hence $a = 4$ and $a {r}^{4} = 324$, Dividing latter by former, we get

${r}^{4} = \frac{324}{4} = 81$ and hence $r = \pm 3$

Hence geometric sequence is $\left\{4 , 12 , 36 , 108 , 324\right\}$ or $\left\{4 , - 12 , 36 , - 108 , 324\right\}$

and middle terms are $\left\{12 , 36 , 108\right\}$ or $\left\{- 12 , 36 , - 108\right\}$