Chapter 7.5, Problem 45PS

Solution

To state: The recursive rule for the balance $a_n$ of the account at the beginning of the nth month if Gladys owes $2000 to a credit card company that charges interest at a rate of 1.4% per month. At the end of each month she makes a payment of $100.

The recursive rule is $a_1=2000,a_n=1.014a_{n-1}-100$ .

Given information:

Gladys owes $ 2000 to a credit card company that charges interest at a rate of 1.4 % per month. At the end of each month she makes a payment of $ 100.

Explanation:

Initially the balance was $2000. Then $a_0=2000$ .

Gladys makes a payment of $ 100 at the end of each month.

Then the balance after a month is:

  $\begin{array}{c}{a}_1=2000+2000\left(\frac{1.4}{100}\right)-100\\ =2000+2000\left(0.014\right)-100\end{array}$

It can be expressed as: $a_1=a_0\left(1.014\right)-100$

Then the balance after two month is:

  $\begin{array}{c}{a}_2=\left(1.014\right)\left[2000\left(1.014\right)-100\right]-100\\ =\left(1.014\right)a_1-100\end{array}$

The balance after n months given by:

  $a_n=1.014a_{n-1}-100$

The recursive rule for the balance after n months is given by:

  $a_1=2000,a_n=1.014a_{n-1}-100$