Chapter 8.1, Problem 16E

(a)

To determine

To find the common ratio.

Answer to Problem 16E

The common ratio is $1.5$

Explanation of Solution

Given:

Sequence, $1,1.5,2.25,3.375,\dots$

Concept Used:

A sequence $\left\{a_n\right\}$ is a geometric sequence if it can be written in the form $\left\{a,a\cdot r,a\cdot r^2,\dots ,a\cdot r^{n-1},\dots \right\}$ for some constant $r$ ,the number $r$is the common ratio.

Calculation:

By comparing the geometric sequence:

  $\begin{array}{c}a\cdot r=1.5,\\ a=1\end{array}$

The common ratio is:

  $\begin{array}{c}r=\frac{1.5}{1}\\ =1.5\end{array}$

Conclusion:

The common ratio is $1.5$

(b)

To determine

The ninth term.

Answer to Problem 16E

The ninth term is $25.6289$

Explanation of Solution

Given:

Sequence, $1,1.5,2.25,3.375,\dots$

Concept Used:

The $n^{th}$ term is:

  $a_n=a\cdot r^{n-1}\text{for}\text{all n}\ge \text{2}$

Where, $a$ =First term, $r$ =common ratio and $n$ = $n^{th}$ term.

Calculation:

Here,

  $a=1$ ,

  $r=1.5$

Ninth term is:

  $\begin{array}{c}{a}_9=1\cdot {\left(1.5\right)}^8\\ =25.6289\end{array}$

Conclusion:

The ninth term is $25.6289$ .

(c).

To determine

To find a recursive rule for $n^{th}$ term.

Answer to Problem 16E

A recursive rule for $n^{th}$ term.is $a_n=\left(1.5\right)\cdot a_{n-1}$ .

Explanation of Solution

Given:

Sequence, $1,1.5,2.25,3.375,\dots$

Concept Used:

Each term is a geometric sequence can be obtained recursively from its preceding term by multiplying $r$ :

  $a_n=a_{n-1}\cdot r\text{for}\text{all}n\ge 2$

Calculation:

Here,

  $a=1$ ,

  $r=1.5$

By using recursive rule, $n^{th}$ is:

  $\begin{array}{c}{a}_n=r\cdot a_{n-1}\\ =\left(1.5\right)\cdot a_{n-1}\end{array}$

Conclusion:

A recursive rule for $n^{th}$ term.is $a_n=\left(1.5\right)\cdot a_{n-1}$ .

(d).

To determine

To find an explicit rule for $n^{th}$ term.

Answer to Problem 16E

An explicit rule for $n^{th}$ term is $a_n=8\cdot {\left(1.5\right)}^{n-1}$ .

Explanation of Solution

Given:

Sequence, $1,1.5,2.25,3.375,\dots$

Concept Used:

The rule for explicit $n^{th}$ is:

  $a_n=a\cdot {\left(r\right)}^{n-1}$

Where, $a$ =First term, $r$ =common ratio and $n$ = $n^{th}$ term.

Calculation:

Here,

  $a=1$ ,

  $r=1.5$

The explicit rule is:

  $\begin{array}{c}{a}_n=a\cdot r^{n-1}\\ =8\cdot {\left(1.5\right)}^{n-1}\end{array}$

Conclusion:

An explicit rule for $n^{th}$ term is $a_n=8\cdot {\left(1.5\right)}^{n-1}$ .