Chapter 7.8, Problem 21PPE

To determine

Tofind:The common ratio of thegiven geometric sequence.

Answer to Problem 21PPE

The common ratio of the geometric sequence $2,-6,18,-54,\dots$ is $-3$ .

Explanation of Solution

Given information:

The given geometric sequence is $2,-6,18,-54,\dots$ .

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 commonratio, 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 geometric sequence is $\langle s\rangle =2,-6,18,-54,\dots$ , that may be written as

  $\langle s\rangle =2,2\times \left(-3\right),2\times {\left(-3\right)}^2,2\times {\left(-3\right)}^3,\dots$ .

Thus, the common ratio of the sequence $\langle s\rangle =2,-6,18,-54,\dots$ is $r=-3$ .

The ratios of consecutive terms are.

  $\frac{-6}{2}=-3,\frac{18}{-6}=-3,\frac{-54}{18}=-3,\dots$

The ratios of consecutive terms for this sequence is constant $-3$ .

Hence, the common ratio of the geometric sequence $\langle s\rangle =2,-6,18,-54,\dots$ is $r=-3$ .

Conclusion:

The common ratio of the geometric sequence $\langle s\rangle =2,-6,18,-54,\dots$ is $-3$ .