Chapter 8.3, Problem 42E

To determine

To find: an expression for nth term and the 22^nd term of the geometric sequence.

Answer to Problem 42E

An expression for nth term is $a_n=4{\left(2\right)}^{n-1}$ .

The 22^ndterm of the geometric sequence will be $a_{22}=8388608$ .

Explanation of Solution

Given information:

Three terms of the sequence are given by,

  $4,8,16,\mathrm{...}$

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.

  $4,8,16,\mathrm{...}$

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

  $\begin{array}{l}8=4r\\ r=\frac{8}{4}\\ r=\mathrm{2...............}(1)\end{array}$

Now, obtain the expression for nth term as shown:

  $\begin{array}{c}{a}_n=a_1{r}^{n-1}\\ =4\times {\left(2\right)}^{n-1}\end{array}$

Thus, an expression for nth term is $a_n=4{\left(2\right)}^{n-1}$ .

Now, calculate the value of $a_{22}$ by using the formula as shown:

  $\begin{array}{c}{a}_{22}=a_1{r}^{22-1}\\ a_{22}=a_1{r}^{21}\\ a_{22}=4\times {\left(2\right)}^{21}\\ a_{22}=8388608\end{array}$

Hence, the 22^ndterm will be $a_{22}=8388608$ .