Chapter 12.2, Problem 47E

a.

To determine

To calculate:

Balance of account at the end of month.

Answer to Problem 47E

Balance of account at the end of month is $\$25.05$ .

Explanation of Solution

Given information:

Tonisha will need $\$750$ to purchase a computer system that will meet her needs.

Calculation:

For avoiding purchasing the computer on credit, she opens a savings account earning of APR of $2.4\%$ compounded monthly and deposits $\$25$ at the beginning of each month.

The interest is compounded each month.

So $n=12$ and interest rate per period is,

  $\frac{\left(2.4\%\right)}{12}=0.2\%$

The common ratio $r$ for geometric series then,

  $\begin{array}{l}r=1+0.002\\ r=1.002\end{array}$

First term $a_1$ in this series is the account balance at the end of the month.

Therefore, the balance of account at the end of first month is,

  $\begin{array}{l}{a}_1=\left(\$25\right)\left(1.002\right)\\ a_1=\$25.05\end{array}$

Thus, Balance of account at the end of month is $\$25.05$ .

b.

To determine

To explain:

Total savings of tonisha if she continues deposit schedule for next two years.

Answer to Problem 47E

Total savings of tonisha if she continues deposit schedule for next two years is $\$615.23$ so she will not able to buy computer.

Explanation of Solution

Apply the formula for the sum of a geometric series,

  $S_n=\frac{a_1-a_1{r}^n}{1-r}$

For end of two years $n=24$ Total savings of her is

  $\begin{array}{l}{S}_n=\frac{\left(\$25.05\right)-\left(\$25.05\right){\left(1.002\right)}^{24}}{1-1.002}\\ S_n=\$615.23\end{array}$

Thus, Total savings of tonisha if she continues deposit schedule for next two years is $\$615.23$ so she will not able to buy computer.

c.

To determine

To calculate:

Least amount of money tonisha can deposit each month.

Answer to Problem 47E

Least amount of money tonisha can deposit each month is $\$30.54$ .

Explanation of Solution

Given information:

Tonisha will need $\$750$ to purchase a computer system that will meet her needs.

Calculation:

It is given that $S=\$750$ .

To find the least amount one apply sum of the geometric series.

Isolate the $a_1$ from the geometric series formula.

  $\begin{array}{l}{S}_n=\frac{a_1-a_1{r}^n}{1-r}\\ a_1=\frac{\left(S_n\right)-\left(1-r\right)}{1-r^n}\end{array}$

On substituting the given values in above equation, least amount is,

  $\begin{array}{l}{a}_1=\frac{\left(\$750\right)-\left(1-1.002\right)}{1-{\left(1.002\right)}^n}\\ a_1=\$30.54\end{array}$

Thus, Least amount of money tonisha can deposit each month is $\$30.54$ .