Chapter 7, Problem 20TP

Solution

i.

To identify: A recursive rule for the balance $a_n$ of the loan at the beginning of the nth month. If Mark takes out a loan for $16,000 with an interest rate of 0.75% per month. At the end of each month, he makes a payment of $300.

The required rule is $a_1=16000,a_n=1.0075a_{n-1}-300,n\ge 2$ .

Given information:

Mark takes out a loan for $16,000 with an interest rate of 0.75% per month. At the end of each month, he makes a payment of $300.

Explanation:

Consider the given information.

The monthly interest rate is 0.75%.

The loan amount was $16,000 and in the second month the monetary debt will increase by $0.75\%$ and amount paid is $300.

  $\begin{array}{c}{a}_2=\left(1+0.75\%\right)16000-300\\ =\left(1.0075\right)16000-300\end{array}$

So, the recursive rule can be $a_1=16000,a_n=1.0075a_{n-1}-300,n\ge 2$

ii.

To calculate: The amount owes at the beginning of the 18th months.

The amount owe is $12749.33 approximately.

Given information:

Mark takes out a loan for $16,000 with an interest rate of 0.75% per month. At the end of each month, he makes a payment of $300.

Calculation:

Consider the given information.

Step 1: Enter the value of $a_1=16000$ into the cell A1.

Step 2: Enter the formula “=1.0075*A1-300” into cell A2 and fill down command to copy the recursive equation into the rest of column A.

  

The amount owe is $12749.33 approximately.

iii.

To calculate: The time taken by Mark to pay off the loan.

69 months are required to pay the loan.

Given information:

Mark takes out a loan for $16,000 with an interest rate of 0.75% per month. At the end of each month, he makes a payment of $300.

Calculation:

Consider the given information.

The recursive rule was $a_1=16000,a_n=1.0075a_{n-1}-300,n\ge 2$

Step 1: Enter the value of $a_1=16000$ into the cell A1.

Step 2: Enter the formula “=1.0075*A1-300” into cell A2 and fill down command to copy the recursive equation into the rest of column A.

  

Thus, it required 69 months to repay the loan because the first term less than 0 is $a_{70}\approx -190$ .

iv.

To Calculate: How long will it take him to pay off the loan if he pays $350 instead of $300 each month and will he end up paying less overall.

It will take 57 months to pay off the loan and it will end up paying less overall.

Given information:

Mark takes out a loan for $16,000 with an interest rate of 0.75% per month. At the end of each month, he makes a payment of $300.

Calculation:

Consider the given information.

For $350 the recursive rule is $a_1=16000,a_n=1.0075a_{n-1}-350,n\ge 2$

Step 1: Enter the value of $a_1=16000$ into the cell A1.

Step 2: Enter the formula “=1.0075*A1-300” into cell A2 and fill down command to copy the recursive equation into the rest of column A.

  

Thus, it required 57 months to repay the loan because the first term less than 0 is $a_{58}\approx -283$ .

It is beneficial to pay the additional $50 because the monthly rate increases the det each month.

If $50 extra will be paid then they need to pay less interest and will repay it faster and have less increases.