Chapter 8, Problem 2CA

a.

To determine

To write a recursive rule for the balance a_n of the loan at the beginning of the nth month.

Answer to Problem 2CA

  $a_n=(1.0075)a_{n-1}-300$

Explanation of Solution

Given:

You lake out a loan for $16,000 with an interest rate of 0.75% per month. At the end of each month, you make a payment of $300.

Calculation:

  $\begin{array}{l}{a}_1=16000\\ a_n=(1.0075)a_{n-1}-300\end{array}$

b.

To determine

To find how much do you owe at the beginning of the 18th month?

Answer to Problem 2CA

  $12749$

Explanation of Solution

Given:

You lake out a loan for $16,000 with an interest rate of 0.75% per month. At the end of each month, you make a payment of $300.

Calculation:

Substituting the value of $n=18$ in the following sequence as follows-

  $\begin{array}{l}{a}_n=(1.0075)a_{n-1}-300\\ a_{18}=12749\end{array}$

c.

To determine

To find how long will it take to pay off the loan?

Answer to Problem 2CA

  ${70}^{th}$ Month.

Explanation of Solution

Given:

You lake out a loan for $16,000 with an interest rate of 0.75% per month. At the end of each month, you make a payment of $300.

Calculation:

  

After ${70}^{th}$ month it finishes.

d.

To determine

To explain if you pay $350 instead of $300 each month, how long will it take to pay off the loan? How much money will you save?

Answer to Problem 2CA

  ${58}^{th}$ Month.

Explanation of Solution

Given:

You lake out a loan for $16,000 with an interest rate of 0.75% per month. At the end of each month, you make a payment of $300.

Calculation:

  $\begin{array}{l}{a}_1=16000\\ a_n=(1.0075)a_{n-1}-350\end{array}$

  

After ${58}^{th}$ month it finishes.