Question

How do you find the geometric means in the sequence 4 , _ , _ , _ , 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 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 .