Chapter 11, Problem 7STP

(a)

To determine

To find: the common ratio of the sequence

Answer to Problem 7STP

Here, the common ratio of the sequence is $4$ .

Explanation of Solution

Given:

The first term of geometric sequence is $3$ .

And the fourth term is $192$ .

Calculation:

The objective is to find the common ratio of the sequence.

The common ratio is calculated below.

  $\begin{array}{l}{a}_4=a_1{r}^{4-1}\\ 192=3r^3\\ 64=r^3\\ 4^3=r^3\\ 4=r\end{array}$

Conclusion:

Hence, the common ratio of the sequence is $4$ .

(b)

To determine

To find: the equation to find the $nth$ term.

Answer to Problem 7STP

Here, the equation to find the $nth$ term is $a_n=a_1{4}^{n-1}$ .

Explanation of Solution

Calculation:

The objective is to write a $nth$ term equation of the sequence.

The $nth$ term of the sequence is written below.

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

Conclusion:

Hence, the equation to find the $nth$ term is $a_n=a_1{4}^{n-1}$ .

(c)

To determine

To find: $6th$ term of the sequence.

Answer to Problem 7STP

Here, the $6th$ term is $3072$ .

Explanation of Solution

Calculation:

The objective is to find the $6th$ term of the sequence.

The $6th$ term of the sequence is calculated below.

  $\begin{array}{l}{a}_n=a_1{r}^{n-1}\\ a_6=3{\left(4\right)}^{6-1}\\ a_6=3{\left(4\right)}^5\\ a_6=3072\end{array}$

Conclusion:

Hence, the $6th$ term is $3072$ .