Chapter 7.8, Problem 12PPE

To determine

Todetermine:whether the given sequence is a geometric sequence.

Answer to Problem 12PPE

The sequence $\langle s\rangle =162,54,18,6,\dots$ is a geometric sequence.

Explanation of Solution

Given information:

The given sequence is $162,54,18,6,\dots$ ,

Conceptused:

A sequence of the form $a,ar,a{r}^2,a{r}^3,\dots ,a{r}^{n-1},\dots$ is the geometric with a as the first term, r the common ratio and nth term $T_n=a{r}^{n-1}$ .

A geometric sequence is a sequence in which the ratio consecutive terms is constant.

The given sequence is $\langle s\rangle =162,54,18,6,\dots$ , that may be written as

  $\langle s\rangle =162,162\times \frac{1}{3}=54,162\times \frac{1}{9}=18,162\times \frac{1}{27}=6,\dots$ .

  $\langle s\rangle =162,162\times \frac{1}{3}=54,54\times \frac{1}{3}=18,18\times \frac{1}{3}=6,\dots$

The ratios of consecutive terms for this sequence is constant, it is $\frac{54}{162}=\frac{1}{3},\frac{18}{54}=\frac{1}{3},\frac{6}{18}=\frac{1}{3},\dots$

Hence, the sequence $\langle s\rangle =162,54,18,6,\dots$ is a geometric sequence.

Conclusion:

The sequence $\langle s\rangle =162,54,18,6,\dots$ is a geometric sequence.