Chapter 9.3, Problem 42E

To determine

To find: the 8^th term of the geometric sequence.

Answer to Problem 42E

The 8^thterm of the geometric sequence will be $a_8=-\frac{2^{24}}{3^{24}}$ .

Explanation of Solution

Given information:

Three terms of the sequence are given by,

  ${\left(\frac{3}{2}\right)}^4,-1,{\left(\frac{2}{3}\right)}^4,\mathrm{...}$

Calculation:

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.

Now $a_2=a_1{r}^{2-1}$

  $\begin{array}{l}-1=\frac{3^4}{2^4}r\\ r=-\frac{2^4}{3^4}\mathrm{...............}(1)\end{array}$

Now, calculate the value of $a_8$ by using the formula as shown:

  $\begin{array}{c}{a}_8=a_1{r}^{8-1}\\ a_8=a_1{r}^7\\ a_8=\frac{3^4}{2^4}\times {\left(-\frac{2^4}{3^4}\right)}^7\\ a_8=-\frac{2^{24}}{3^{24}}\end{array}$

Hence, the 8^thterm will be $a_8=-\frac{2^{24}}{3^{24}}$ .