Chapter 7.8, Problem 20PPE

To determine

Tofind:The common ratio of thegiven geometric sequence.

Answer to Problem 20PPE

The common ratio of the geometric sequence $7,-7,7,-7,\dots$ is $-1$ .

Explanation of Solution

Given information:

The given geometric sequence is $7,-7,7,-7,\dots$ .

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 geometric sequence is $\langle s\rangle =7,-7,7,-7,\dots$ , that may be written as

  $\langle s\rangle =7,7\times \left(-1\right),7\times {\left(-1\right)}^2,7\times {\left(-1\right)}^3,\dots$ .

Thus, the common ratio of the sequence $\langle s\rangle =7,-7,7,-7,\dots$ is $r=-1$ .

The ratios of consecutive terms are.

  $\frac{-7}{7}=-1,\frac{7}{-7}=-1,\frac{-7}{7}=-1,\dots$

The ratios of consecutive terms for this sequence is constant $-1$ .

Hence, the common ratio of the geometric sequence $\langle s\rangle =7,-7,7,-7,\dots$ is $r=-1$ .

Conclusion:

The common ratio of the geometric sequence $\langle s\rangle =7,-7,7,-7,\dots$ is $-1$ .