This problem aims to familiarize us with **missing numbers** in different sets of **series.** The concept required to solve the given problem is basic **calculus** involving **sequences** and **series.**

**Sequence** and **series** are the basic topics of **arithmetic’s.** We define a **sequence** as an enumerated group of numbers or elements in which **recurrences** of any sort are permitted, whereas a **series** is the **sum** of all **numbers** or elements

Whereas the **numbers** that are **skipped** in the given series of a number with **identical** differences among them are known as **missing numbers** in the series. The **technique** of finding the missing numbers is **defined** as figuring out the similar changes between those numbers and loading the missing number in the distinctive **series** and **places.**

## Expert Answer

Here, we are given a **geometric sequence,** in which every **element** is acquired by **multiplying** or **dividing** a definite numeral with the initial number. The **steps** to find the missing number are:

**-Choose**$2$ or $3$ numbers to which the rule will be used to**uncover**the missing number. Let’s say you have $5$**numbers**in a**series,**choose the first $3$**elements**to match the**rule**that is to be used.**–**While selecting the**number**to match the**rule,**pick the number that is**effortless**to**work**with. These contain numbers that are**factors**of $2,3,5$ or $10$. You can also review the**series**with some**familiar**forms such as**squares, cubes,**etc.

The given **series** is:

\[9,\space ?,\space 6561,\space 43046721\]

We have to **determine** the number $?$ in the series.

So by looking at the **series,** we can infer that the $3rd$ and the $4th$ **numbers** have some **connection **and if we find this **connection,** we can acquire the relation of the **whole series** and thus find the **missing number.** So finding the **relation** between $6561$ and $43046721$.

If we **multiply** the $3rd$ number by itself it **produces** the $4th$ number:

\[6561\times 6561=43046721\]

So by this, we can say that each **number** in the series is the square of the **previous number.**

\[a_{n}=(a_{n-1})^2\]

So to find $2nd$ **number,** inserting $n=2$:

\[a_{2}=(a_{2-1})^2 \]

\[a_{2} =(a_{1})^2 \]

\[a_{2} = (9)^2 \]

That is:

\[a_{2} = 81\]

For **confirmation** let’s now produce the 3rd number $a_3$ using the $2nd$ number $a_2$ and see if the **relation** for the **series** is correct.

\[a_{3} = (a_{3-1})^2\]

\[a_{3} = (a_{2})^2\]

\[a_{3} = (81)^2\]

\[a_{3} = 6561\]

So the missing term is **confirmed** to be $81$.

## Numerical Result

The **missing number** in the series $9, \space ? \space, \space 6561, \space 43046721$ is $81$.

**Complete** series is:

$9, \space 81, \space 6561, \space 43046721$

## Example

Find the **Missing Number** in series $2, \space 8, \space ?, 134217728$.

By looking at the **series** we can conclude that the **relation** of the series can be found if we find out the **relation** between $2$ and $8$.

The **relationship **is:

\[a_{n} = (a_{n-1})^3\]

So to find $3rd$ number, **inserting** $n=3$:

\[a_{3} = (a_{3-1})^3\]

\[a_{3} = (a_{2})^3\]

\[a_{3} = (8)^3\]

That is:

\[a_{3} = 512\]