Problem 81 - Algebra and Trigonometry

Question

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 \(B_{0}=\$ 3000 \quad B_{n}=1.01 B_{n-1}-100\)

Step-by-Step Solution

Answer

For the first three months, John's balances will be \(\$2930,\ \$2859.30,\ \$2787.893\). Continue using \[ B_{n}=1.01 B_{n-1}-100 \] until the balance is zero.

Step 1: Understand the Problem

John has a credit card balance of \(\$3000\) with a monthly interest rate of \(1\%\). He can afford to pay \(\$100\) each month. The balance each month is given by the sequence \[B_{0}=\$3000 \quad B_{n}=1.01 B_{n-1}-100\].

Step 2: Calculate the Balance for the First Month

To find \(B_{1}\), use the formula with \(B_{0} = 3000\). \[ B_{1} = 1.01 \times 3000 - 100 = 1.01 \times 3000 - 100 = 3030 - 100 = 2930 \]

Step 3: Calculate the Balance for the Second Month

To find \(B_{2}\), use the formula with \(B_{1} = 2930\). \[ B_{2} = 1.01 \times 2930 - 100 = 1.01 \times 2930 - 100 = 2959.30 - 100 = 2859.30 \]

Step 4: Calculate the Balance for the Third Month

To find \(B_{3}\), use the formula with \(B_{2} = 2859.30\). \[ B_{3} = 1.01 \times 2859.30 - 100 = 1.01 \times 2859.30 - 100 = 2887.893 - 100 = 2787.893 \]

Step 5: Generalize the Calculation

Each month, the balance can be calculated using the formula: \[ B_{n} = 1.01 B_{n-1} - 100 \] Repeat this calculation until the balance is near zero.

Key Concepts

Credit Card Debt

Credit card debt can quickly become overwhelming due to interest rates added to the original balance. When you carry a balance from month to month, credit card companies charge interest on the remaining unpaid balance. In the exercise, John has a balance of \(3000 on his Discover card, with a monthly interest rate of 1%. He aims to pay down this debt with regular payments of \)100 each month. To fully grasp his financial commitment, we use a recursive sequence to model his remaining balance month by month.

Interest Rate Calculations

Interest rate calculations are crucial in understanding how debt increases over time. In this exercise, the interest rate of 1% per month means John's balance increases by 1% every month. If John makes payments, the new balance for the next month is the remaining amount after applying the interest and subtracting his payment.
For example:

• Initial balance: \(3000
• Interest applied: 1% of \)3000 = \(30
• New Balance (with interest): \)3030
• Payment made: \(100
• Remaining Balance: \)2930

This pattern repeats each month, showing how interest and payments interact to affect the balance.

Financial Mathematics

Financial mathematics helps us understand and manage money more effectively. In this example, it's applied to calculate how long it takes to pay off debt. The key aspects here include:

• Principal Amount: The original amount of debt, which is \(3000.
• Interest Rate: The rate at which interest accumulates, which is 1% per month in John's case.
• Monthly Payments: The consistent amount John can pay, which is \)100 each month.

Using financial mathematics, one can predict when the debt will be cleared or how adjustments in payments or interest rates can affect the time needed to pay off the debt.
In John's case, the recursive sequence helps model this over multiple months.

Recursively Defined Sequence

A recursive sequence is a sequence of numbers where each term is defined based on the terms before it. In John's debt scenario, the balance each month is determined by the previous month's balance. The formula used is:

• Initial Balance: \(B_{0} = 3000\)
• Recursive Formula: \(B_{n} = 1.01B_{n-1} - 100\)

Here:

• \(B_{n}\) represents the balance in month n.
• 1.01 accounts for the 1% interest.
• -100 accounts for John's monthly payment.

Using this, we can compute the balance for each month step-by-step until it reaches zero.
This recursive approach encapsulates the sequence of payments and interest applications, making it easier to foresee the debt reduction over time.