Chapter 7.8, Problem 18PPE

To determine

Tofind:The common ratio of thegiven geometric sequence.

Answer to Problem 18PPE

The common ratio of the geometric sequence $5,20,80,320,\dots$ is $4$ .

Explanation of Solution

Given information:

The given geometric sequence is $128,96,72,54,\dots$ .

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 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 $\langle s\rangle =128,96,72,54,\dots$ , that may be written as

  $\langle s\rangle =128,128\times \left(\frac{3}{4}\right),128\times {\left(\frac{3}{4}\right)}^2,128\times {\left(\frac{3}{4}\right)}^3,\dots$ .

Thus, the common ratio of the sequence $\langle s\rangle =128,96,72,54,\dots$ is $r=\frac{3}{4}$ .

The ratios of consecutive terms are.

  $\frac{96}{128}=\frac{3}{4},\frac{72}{96}=\frac{3}{4},\frac{54}{72}=\frac{3}{4},\dots$

The ratios of consecutive terms for this sequence is constant $\frac{3}{4}$ .

Hence, the common ratio of the geometric sequence $\langle s\rangle =128,96,72,54,\dots$ is $r=\frac{3}{4}$ .

Conclusion:

The common ratio of the geometric sequence $\langle s\rangle =128,96,72,54,\dots$ is $\frac{3}{4}$ .