Chapter 11.3, Problem 13WE

To determine

To calculate:

The value of $t_5$ for following geometric sequence:

  $t_2=18,t_3=12$

Answer to Problem 13WE

The value of $t_5$ the geometric sequence is $\frac{16}{3}$ .

Explanation of Solution

Given information:

  $t_2=18,t_3=12$

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:

  $\begin{array}{l}{t}_2=t_1{r}^{2-1}\\ 18=t_{1}r\end{array}$

  $t_{1}r=18$  ... (1)

And

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

  $t_1{r}^2=12$  ... (2)

Now, $t_1$ is calculated by taking ratio of both eqns.

  $\begin{array}{l}\frac{t_1{r}^2}{t_{1}r}=\frac{12}{18}\\ r=\frac{2}{3}\end{array}$

Now, put value of $r=\frac{2}{3}$ in eq. (1):

  $\begin{array}{l}{t}_{1}r=18\\ t_1\left(\frac{2}{3}\right)=18\\ t_1=18\left(\frac{3}{2}\right)\\ t_1=27\end{array}$

Now, the value of term $t_5$ is calculated as:

  $\begin{array}{l}{t}_5=t_1{r}^{5-1}\\ t_5=27{\left(\frac{2}{3}\right)}^4\\ t_5=\frac{16}{3}\end{array}$