Chapter 8.3, Problem 27E

To determine

To find: first five terms of the geometric sequence and the common ratio. Write the nth term of the sequence as a function of n .

Answer to Problem 27E

First five terms of the geometric sequence are $9,18,36,72,144$ respectively.

Common ratio of the sequence is $2$ .

An expression for nth term of the sequence is $a_n=9{\left(2\right)}^{n-1}$ .

Explanation of Solution

Given information:

An sequence is given with first term $a_1=9$ and a recurrence relation $a_{k+1}=2a_k$ .

Concept used:

The $n$ th term of geometric sequence has the form $a_1=a_1{r}^{n-1}$ where $r$ is the common ratio of the consecutive terms of the sequence.

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:

Now, consider the first term.

  $a_1=9$

And

  $a_{k+1}=2a_k$

First five terms of the sequence can be found by substituting 1, 2, 3, 4, and 5 for kas shown:

  $\begin{array}{l}{a}_2=2a_1=2\times 9=18\\ a_3=2a_2=2\times 18=36\\ a_4=2a_3=2\times 36=72\\ a_5=2a_4=2\times 72=144\end{array}$

Hence, the first five terms of the sequence are $9,18,36,72,144$ respectively.

The common ratio can be found by dividing first term by second term.

  $\begin{array}{c}r=\frac{a_2}{a_1}\\ =\frac{18}{9}\\ =2\end{array}$

Therefore, the common ratio is $2$ .

Now, the expression for nth term can be found as

  $\begin{array}{c}{a}_n=a_1{r}^{n-1}\\ =9{\left(2\right)}^{n-1}\\ a_n=9{\left(2\right)}^{n-1}\end{array}$

Hence, the expression for nth term of the sequence is $a_n=9{\left(2\right)}^{n-1}$ .