Chapter 6.5, Problem 25E

To determine

To find:An equation for the nth term of the geometric sequence. Then find $a_6$ .

Answer to Problem 25E

The nth term of the geometric sequence is $a_n=2\cdot 4^{n-1}$ and $a_6=2048$ .

Explanation of Solution

Giveninformation:

The sequence is $2,8,32,128,\mathrm{...}$

Formula used:

The nth term of a geometric sequence is given by $a_n=a_1{r}^{n-1}$ , where $a_1$ represents the first term of the sequence and $r$ represents the common ratio.

From the given sequence we can notice that $a_1=2$ and $r=4$ .

Therefore, the nth term of the sequence is

  $\begin{array}{l}{a}_n=a_1{r}^{n-1}\\ \Rightarrow a_n=2\cdot 4^{n-1}\end{array}$

To find $a_6$ , substitute 6 for $n$ .

  $a_6=2\cdot 4^{6-1}=2\cdot 4^5=2048$

Conclusion:

Therefore, the nth term of the geometric sequence is $a_n=2\cdot 4^{n-1}$ and $a_6=2048$ .