Chapter 6.6, Problem 50E

(a)

To determine

The equation that represents the nth term of the geometric sequence.

Answer to Problem 50E

The function is:

  $f\left(n\right)=0.01{\left(2\right)}^{n-1}$

Explanation of Solution

Given information:

She will pay her parents of first month is $0.01

She will pay her parents of second month is $0.02

She will pay her parents of third month is $0.04

Formula used:

The function for a geometric sequence is:

  $f\left(n\right)=a_1{r}^{n-1}$

Calculation:

The function for a geometric sequence is:

  $f\left(n\right)=a_1{r}^{n-1}$

From the given, $a_1=0.01\&r=2$

Therefore, the function is:

  $f\left(n\right)=0.01{\left(2\right)}^{n-1}$

Conclusion:

The function is:

  $f\left(n\right)=0.01{\left(2\right)}^{n-1}$

(b)

To determine

The amount that she will pay on 25^th day.

Answer to Problem 50E

She will have to pay $167772.16 on the 25^th day

Explanation of Solution

Given information:

She will pay her parents of first month is $0.01

She will pay her parents of second month is $0.02

She will pay her parents of third month is $0.04

Formula used:

The function for a geometric sequence is:

  $f\left(n\right)=a_1{r}^{n-1}$

Calculation:

At $n=25$

  $\begin{array}{l}f\left(25\right)=0.01{\left(2\right)}^{25-1}\\ f\left(25\right)=0.01{\left(2\right)}^{24}\\ f\left(25\right)=167772.16\end{array}$

She will have to pay $167772.16 on the 25^th day

Conclusion:

She will have to pay $167772.16 on the 25^th day

(c)

To determine

if the student had made a good choice or not.

Answer to Problem 50E

She should have chosen to live on campus because of the payments she will have to make living with her parents.

Explanation of Solution

Given information:

She will pay her parents of first month is $0.01

She will pay her parents of second month is $0.02

She will pay her parents of third month is $0.04

Formula used:

The function for a geometric sequence is:

  $f\left(n\right)=a_1{r}^{n-1}$

Calculation:

She should have chosen to live on campus because of the payments she will have to make living with her parents. Imagine she will be pay more than $5 million on the 30th day!

Conclusion:

She should have chosen to live on campus because of the payments she will have to make living with her parents.