Chapter 8.3, Problem 39E

To determine

To find: an expression for nth term and the 9^h term of the geometric sequence.

Answer to Problem 39E

An expression for nth termis $a_n=7{\left(3\right)}^{n-1}$ .

The 9^thterm of the geometric sequence will be $a_9=45927$ .

Explanation of Solution

Given information:

Three terms of the sequence are given by,

  $7,21,63,\mathrm{...}$

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:

Consider the given sequence.

  $7,21,63,\mathrm{...}$

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

  $\begin{array}{l}21=7r\\ r=\frac{21}{7}\\ r=\mathrm{3...............}(1)\end{array}$

Now, obtain the expression for nth term as shown:

  $\begin{array}{c}{a}_n=a_1{r}^{n-1}\\ =7\times {\left(3\right)}^{n-1}\end{array}$

Thus, an expression for nth term is $a_n=7{\left(3\right)}^{n-1}$ .

Now, calculate the value of $a_9$ by using the formula as shown:

  $\begin{array}{c}{a}_9=a_1{r}^{9-1}\\ a_9=a_1{r}^8\\ a_9=7\times {\left(3\right)}^8\\ a_9=45927\end{array}$

Hence, the 9^thterm will be $a_9=45927$ .