Chapter 6.5, Problem 50E

a.

To determine

To write an equation that represents the $n$ th term of the geometric sequence.

Answer to Problem 50E

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

Explanation of Solution

Given:

A college student makes a deal with her parents to live at home instead of living on campus. She will pay her parents $\$0.01$ for the first day of the month, $\$0.02$ for the second day, $\$0.04$ for the third day, and so on.

The function for a geometric sequence is:

  $f(n)=a_1{r}^{n-1}$

From the given,

  $a_1=0.01$ and $r=2$

Hence,

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

b.

To determine

To find the amount paid by her on the 25^th day.

Answer to Problem 50E

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

Explanation of Solution

Given:

A college student makes a deal with her parents to live at home instead of living on campus. She will pay her parents $\$0.01$ for the first day of the month, $\$0.02$ for the second day, $\$0.04$ for the third day, and so on.

  $n=25$

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

Hence,

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

c.

To determine

To explain about the student, make a good choice or should she have chosen to live on campus.

Answer to Problem 50E

The payments she will have to pay more than $\$5$ on the $30th$ day to make living with her parents.

Hence,

She should have chosen to live on campus.

Explanation of Solution

Given:

A college student makes a deal with her parents to live at home instead of living on campus. She will pay her parents $\$0.01$ for the first day of the month, $\$0.02$ for the second day, $\$0.04$ for the third day, and so on.

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