Chapter 11.5, Problem 7CYU

(a)

To determine

Write a recursive formula for the balance owed at the end of each month.

Answer to Problem 7CYU

The recursive formula: $a_{n+1}=1.01·a_n-100;a_1=1500$

Explanation of Solution

Given:

Ben financed a $1500 rowing machine to help him train for the college rowing team. He could only make a $100 payment each month, and his bill increased by 1 % due to interest at the end of each month.

Write a recursive formula for the balance owed at the end of each month.

Concept Used:

Ben financed a $1500 rowing machine to help him train for the college rowing team. He could only make a $100 payment each month, and his bill increased by 1 % due to interest at the end of each month.

According to the condition given in the question:

The recursive formula: $a_{n+1}=1.01·a_n-100;a_1=1500$

Thus, the recursive formula: $a_{n+1}=1.01·a_n-100;a_1=1500$

(b)

To determine

Find the balance owed after the first four months.

Answer to Problem 7CYU

The next 4 balances are: $\$1415;\$1329.15;\$1242.44;\$1154.87$

Explanation of Solution

Given:

Ben financed a $1500 rowing machine to help him train for the college rowing team. He could only make a $100 payment each month, and his bill increased by 1 % due to interest at the end of each month.

Find the balance owed after the first four months.

Concept Used:

First month balance is $1500.

Using the recursive formula $a_{n+1}=1.01·a_n-100;a_1=1500$ we can find the balance owed in next 4 months.

Calculation:

  $a_n=1.01·a_{n-1}-100;a_1=1500$

The balance owed after the first four months. $\begin{array}{l}{a}_2=1.01·a_1-100=1.01·1500-100=1415\\ a_3=1.01·a_2-100=1.01·1415-100=1329.15\\ a_4=1.01·a_3-100=1.01·1329.15-100=1242.44\\ a_5=1.01·a_4-100=1.01·1242.44-100=1154.87\end{array}$

Thus, the next 4 balances are: $\$1415;\$1329.15;\$1242.44;\$1154.87$

(c)

To determine

Find the amount of interest has accumulated after the first six months.

Answer to Problem 7CYU

$77.08

Explanation of Solution

Given:

Ben financed a $1500 rowing machine to help him train for the college rowing team. He could only make a $100 payment each month, and his bill increased by 1 % due to interest at the end of each month.

How much interest has accumulated after the first six months?

Concept Used:

Find the balance owed after the first six months.

Calculation:

Find the balance owed after the first six months.

  $\begin{array}{l}{a}_6=1.01·a_5-100=1.01·1154.87-100=1066.42\\ a_7=1.01·a_6-100=1.01·1066.42-100=977.84\end{array}$

Amount paid for the first six months is $600.

Remaining amount = $\$(1500-600)=\$900$

So, after six months $900 should be paid if there is no interest at all. $977.84–900=77.08$

After the first six months, the interest has accumulated to $77.08.

Thus, after the first six months, the interest has accumulated to $77.08