Chapter 12.1, Problem 81AYU

Credit Card Debt John has a balance of $\$3000$ on his Discover card, which charges $1\%$ interest per month on any unpaid balance from the previous month. John can afford to pay $100 toward the balance each month. His balance each month after making a $\$100$ payment is given by the recursively defined sequence

$B_0=\$3000B_n=1.01B_{n-1}-100$.

Determine John's balance after making the first payment. That is, determine $B_1$.

To determine

John’s balance after making the first payment. That is, $B_1$ where, John has a balance of $\$3000$ on his Discover card, which charges $1\%$ interest per month on any unpaid balance from the previous month. John can afford to pay $\$100$ toward the balance each month. His balance each month after making a $\$100$ payment is defined by recursive sequence: $B_0=\$3000B_n=1.01B_{n-1}-100$.

Answer to Problem 81AYU

Solution:

John’s balance is $\$2930$ after making the first payment.

Explanation of Solution

Given information:

John has a balance of $\$3000$ on his Discover card, which charges $1\%$ interest per month on any unpaid balance from the previous month. John can afford to pay $\$100$ toward the balance each month. His balance each month after making a $\$100$ payment is defined by recursive sequence: $B_0=\$3000B_n=1.01B_{n-1}-100$.

Explanation:

Consider the recursive relation, $B_0=\$3000B_n=1.01B_{n-1}-100$.

To find John’s balance after making the first payment. That is, $B_1$ substitute $n=1$ in

$B_n=1.01B_{n-1}-100$.

$\Rightarrow B_1=1.01B_{1-1}-100$

$\Rightarrow B_1=1.01B_0-100$

Substitute $B_0=\$3000$ in above equation.

$\Rightarrow B_1=1.01\left(3000\right)-100=2930$.

Therefore, John’s balance after making the first payment is, $B_1=\$2930$.