Chapter 9.3, Problem 21E

To determine

To write: the first five terms of the geometric sequence.

Answer to Problem 21E

The first five terms of the geometric sequence are $2,6x,18x^2,54x^3,162x^4$ .

Explanation of Solution

Given information:

The given sequence,

  $a_1=2,r=3x$

Concept used:

The $n$ th term of geometric sequence has the form $a_1=a_1{r}^{n-1}$ where $r$ is the common ratio of the consecutive terms of the sequence.

Every geometric sequence can be written in the following form.

  $a_1=a_1,a_2=a_{1}r,a_3=a_1{r}^2,a_4=a_1{r}^3,\mathrm{...},a_n=a_1{r}^{n-1},\mathrm{..}$

Calculation:

Consider the first term $a_1=2$ and common ratio $r=3x$ :

The first term is $a_1=2$ .

To find the second term, multiply first term by $3x$ a shown below.

  $\begin{array}{c}{a}_2=a_{1}r\\ =2\cdot 3x\\ =6x\end{array}$

So, the second term is $6x$ .

The 3^rd term can be calculated as

  $\begin{array}{c}{a}_3=a_{2}r\\ =6x\cdot 3x\\ =18x^2\end{array}$

So, the third term is $18x^2$

The 4^th term can be calculated as

  $\begin{array}{c}{a}_4=a_{3}r\\ =18x^2\cdot 3x\\ =54x^3\end{array}$

So, the fourth term is $54x^3$ .

The 5^th term can be calculated as

  $\begin{array}{c}{a}_5=a_{4}r\\ =54x^3\cdot 3x\\ =162x^4\end{array}$

So, the fifth term is $162x^4$

Therefore, the five terms of sequence are $2,6x,18x^2,54x^3,162x^4$ .