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**Hint:**In order to find the solution to this question , we need to first understand the mathematical concept of the geometric sequence . A Geometric sequence somewhere also called a geometric progression .

It is actually a sequence formed of non – zero numbers such that there in the sequence

Each term subsequent to the first term goes to the next by always multiplying by the same , fixed non – zero number called the common ratio and denoted by ‘ r ‘ . If there are n terms in the sequence then the first term is always denoted by ‘ ${a_1}$ ’ .

**Complete Step by step solution :**

According to the given question , our first term is ${a_1} = 4$. Also the common ratio as described in the hint part can be calculated by dividing any number or term from the sequence by the term preceding it .

Now The first term is ${a_1} = 4$ and another given term is 324 positioned at fifth place that can be expressed as the term in $a{r^4}$ . So , $a{r^4} = 324$. We can also determine the position of the term by keeping in mind that $a$is the first term , $ar$is the second term , $a{r^2}$is the third term and so on .

So , the common ratio , ‘ r ‘ can be calculated as –

$\dfrac{{a{r^4}}}{a}$= $\dfrac{{324}}{4}$

So here , ${r^4} = 81$

We can easily see that 81 comes when 3 is multiplied four times that is ${3^4} = 81$, comparing this with

${r^4} = 81$ , we get the common ratio , ${r^{}} = \pm 3$ .

So , the common ratio can be 3 and -3 .

Accordingly we will make the geometric sequence using the common ratio as 3 and as well as -3 .

Hence , the geometric sequence generated is \[\{ 4,12,36,108,324\} \]o r \[\{ 4, - 12, - 36, - 108, -

324\} \].

**So, the middle terms are \[\{ 12,36,108\} \]or \[\{ - 12, - 36, - 108\} \].**

**Note :**If the same number is not multiplied to each number in the series, then there is no common ratio.

Alternatively , to find the nth term of the sequence is determined by the formula = ${a_n} = a{r^{n -

1}}$.

If the common ratio is determined to be a complex number then also geometric series is said to be valid .

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