Chapter 11.3, Problem 14WE

To determine

To calculate:

The value of $t_9$ for following geometric sequence:

  $t_3=-12,t_6=96$

Answer to Problem 14WE

The value of $t_9$ the geometric sequence is $-768$ .

Explanation of Solution

Given information:

  $t_3=-12,t_6=96$

Formula used:

The $n^{th}$ term of geometric sequence is given by the following expression:

  $t_n=t_1{r}^{n-1}$

Where,

First term: $t_1$

Common difference: $r$

Calculation:

The equations are formed as:

  $\begin{array}{l}{t}_3=t_1{r}^{3-1}\\ -12=t_1{r}^2\end{array}$

  $t_1{r}^2=-12$  ... (1)

And

  $\begin{array}{l}{t}_6=t_1{r}^{6-1}\\ 96=t_1{r}^5\end{array}$

  $t_1{r}^5=96$  ... (2)

Now, $t_1$ is calculated by taking ratio of two equations:

  $\begin{array}{l}\frac{t_1{r}^5}{t_1{r}^2}=\frac{96}{-12}\\ r^3=-8\\ r=-2\end{array}$

Now, put value of $r=-2$ in eq. (1):

  $\begin{array}{l}{t}_1{r}^2=-12\\ t_1{\left(-2\right)}^2=-12\\ t_1=-\frac{12}{4}\\ t_1=-3\end{array}$

Now, the value of term $t_9$ is calculated as:

  $\begin{array}{l}{t}_9=t_1{r}^{9-1}\\ t_9=\left(-3\right){\left(-2\right)}^8\\ t_9=-768\end{array}$