Chapter 7.8, Problem 10PPE

To determine

Todetermine:whether the given sequence is a geometric sequence.

Answer to Problem 10PPE

The sequence $2,8,32,128,\dots$ is a geometric sequence.

Explanation of Solution

Given information:

The given sequence is $2,8,32,128,\dots$ ,

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 $T_n=a{r}^{n-1}$ .

The given sequence is $\langle s\rangle =2,8,32,128,\dots$ , that may be written as

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

Hence, sequence $\langle s\rangle =2,8,32,128,\dots$ is a geometric sequence with 2 as the first term, 4 the common ratio and nth term $T_n=2\times 4^{n-1}$ .

Conclusion:

The sequence $2,8,32,128,\dots$ is a geometric sequencewith 2 as the first term, 4 the common ratio and nth term $T_n=2\times 4^{n-1}$ .