Chapter 11.5, Problem 7P

a.

To determine

To calculate:

Total that plan A will pay.

Answer to Problem 7P

Total that plan A will pay is $\$105$ .

Explanation of Solution

Give information:

Plan A: $\$1$ on first day, $\$2$ on the second day, $\$3$ on third day, and so on for two weeks.

Calculation:

It is given that,

  $\$1$ on first day, $\$2$ on the second day, $\$3$ on third day, and so on for two weeks.

So this is arithmetic series,

  $a_n=a_1+\left(n-1\right)a_n$

Here,

  $\begin{array}{l}{a}_1=1\\ d=\$2-\$1=1\end{array}$

After putting the values in above equation,

  $\begin{array}{l}{a}_n=1+\left(n-1\right)1\\ a_n=n\end{array}$

Formula for sum is,

  $\begin{array}{l}{S}_n=\frac{n}{2}\left(1+n\right)\\ S_{14}=\frac{14}{2}\left(1+14\right)\\ S_{14}=\$105\end{array}$

Therefore, Total that plan A will pay is $\$105$ .

b.

To determine

To calculate:

Total that plan B will pay.

Answer to Problem 7P

Total that plan B will pay is $\$1638.3$ .

Explanation of Solution

Give information:

Plan B: $1\varphi$ on first day, $2\varphi$ on the second day, $4\varphi$ on third day, and so on for two weeks.

Calculation:

It is given that,

  $1\varphi$ on first day, $2\varphi$ on the second day, $4\varphi$ on third day, and so on for two weeks.

Common ratio is calculated as,

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{a_3}{a_2}\\ r=\frac{2}{1}=\frac{4}{2}\\ r=2\end{array}$

So this is geometric series,

Here,

  $\begin{array}{l}{a}_1=0.1\\ r=2\end{array}$

After putting the values in sum equation,

  $\begin{array}{l}{S}_{14}=\frac{a\left(1-r^n\right)}{\left(1-r\right)}\\ S_{14}=\frac{0.1\left(1-2^{14}\right)}{\left(1-2\right)}\\ S_{14}=\$1638.3\end{array}$

Therefore, Total that plan B will pay is $\$1638.3$ .