Chapter 9.3, Problem 25E

Solution

a.

To calculate: The common ratio of the sequence $2,6,18,54,\mathrm{...}$

The common ratio of the sequence is 3.

Given information:

The sequence is $2,6,18,54,\mathrm{...}$

Formula used:

The common ratio ( r ) is $r=\frac{a_2}{a_1}$ where $a_2$ is the second term of the sequence and $a_1$ is the first term of the sequence.

Calculation:

Consider the given sequence.

  $2,6,18,54,\mathrm{...}$

Use the common ratio formula.

  $\begin{array}{c}r=\frac{6}{2}\\ =3\end{array}$

The common ratio is 3.

b.

To calculate: The eighth term of the sequence $2,6,18,54,\mathrm{...}$

The eighth term of the given sequence is 42.

Given information:

The given sequence is $2,6,18,54,\mathrm{...}$

Formula used:

The formula for the nth term of a geometric sequence is $a_n=a_1{r}^{n-1}$ where $a_1$ is the first term n is the nth term and r is the common ratio.

Calculation:

Consider the given sequence.

The common ratio is 3, the first term is 2 and the value of n is 8.

Substitute the respective values into the formula $a_n=a_1{r}^{n-1}$ .

  $\begin{array}{c}{a}_n=2{\left(3\right)}^{8-1}\\ =2{\left(3\right)}^7\\ =4374\end{array}$

The eighth term of the given sequence is 4374.

c.

To identify: The recursive rule for the nth term of the sequence $2,6,18,54,\mathrm{...}$

The recursive rule is $a_1=2,a_n=3a_{n-1}\text{ for }n\ge 2$ .

Given information:

The given sequence is $2,6,18,54,\mathrm{...}$

Explanation:

Consider the given sequence.

Let the first term of the sequence is $a_1$ thus the formula for the nth term will be $a_n=a_{n-1}\left(r\right)\text{ for }n\ge 2$ where r is the common ratio.

Since the first term is 2 and the common ratio is 3 thus,

  $a_1=2,a_n=3a_{n-1}\text{ for }n\ge 2$

d.

To identify: An explicit rule for the nth term of the sequence $2,6,18,54,\mathrm{...}$

The explicit rule is $a_n=2{\left(3\right)}^{n-1}$ .

Given information:

The given sequence is $2,6,18,54,\mathrm{...}$

Explanation:

Consider the given sequence.

It is known that the common ratio is 3, and the first term, $a=2$ .

Substitute the respective values into the formula $a_n=a_1{r}^{n-1}$ .

  $a_n=2{\left(3\right)}^{n-1}$

The explicit rule is $a_n=2{\left(3\right)}^{n-1}$ .