Chapter 8.3, Problem 45E

To determine

To find: the 6th term of the geometric sequence.

Answer to Problem 45E

The 6th term of the geometric series will be $a_6=-\frac{2}{9}$ .

Explanation of Solution

Given information:

Two terms of the sequence,

  $a_2=-18,a_5=\frac{2}{3}$

Concept used:

The general term of geometric sequence is given by formula

  $a_n=a_1{r}^{n-1}$

Here, $a_1$ is the first term and $r$ is common ratio.

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:

Now $a_5=a_2{r}^3$

  $\begin{array}{c}{a}_5=a_2{r}^3\\ -18=\frac{2}{3}{r}^3\\ r=-\frac{1}{3}\mathrm{...............}(1)\end{array}$

Now, calculate the value of $a_6$ by using the formula as shown:

  $\begin{array}{c}{a}_6=a_{5}r\\ a_6=\frac{2}{3}\left(-\frac{1}{3}\right)\\ a_6=-\frac{2}{9}\end{array}$

Hence, 6^thterm will be $a_6=-\frac{2}{9}$ .