Question

How do you write a formula for the general term(the nth term) of the geometric sequence $4,12,36,108,324,...$?

Answer 1

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 as , 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 nth term is represented as ${a_n} = a{r^{n - 1}}$.

Complete step by step solution:
We are given a geometric sequence as $4,12,36,108,324,...$
The general term or the nth term of any geometric sequence is of the form
${a_n} = a{r^{n - 1}}$
Where $a$is the first term of the series and $r$is the factor or the common ratio between the terms.
As we can clearly see that our first term is $a = {a_1} = 4$.
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 $36$ positioned at third place that can be expressed as the term in $a{r^2}$ . So , $a{r^2} = 36$. 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 –
$\dfrac{{a{r^1}}}{a} = \dfrac{{12}}{4}$
So here , ${r^2} = 3$
So , the common ratio can be 3 and
Accordingly we will make the nth term of the geometric sequence using the common ratio as 3. We get
${a_n} = 5{\left( 3 \right)^{n - 1}}$
Therefore, the nth term for the given geometric sequence can be ${a_n} = 5{\left( 3 \right)^{n - 1}}$

Note: If the same number is not multiplied to each number in the series, then there is no common ratio.
Common ratio of any geometric sequence can never be equal to zero.
If the common ratio is determined to be a complex number then also geometric series is said to be valid.