The problem presented in the image involves a credit card balance of $5000 with an APR (Annual Percentage Rate) of 22%, and the minimum payment made each month is 3% of the balance. We need to find a formula for the balance after $t$ monthly payments.

Step-by-step solution:

1. Determine the Monthly Interest Rate:

   • The APR is 22%, so the monthly interest rate, $r$, is: $r = \frac{22\%}{12} = \frac{0.22}{12} \approx 0.01833$

2. Payment Structure:

   • The minimum monthly payment is 3% of the balance, so the payment amount each month is $0.03 \times B(t)$, where $B(t)$ is the balance at month $t$.

3. Recurrence Relation:

   • After making a payment, the remaining balance consists of the previous month's balance minus the payment, plus the interest that has been charged. This can be modeled by the following recursive formula: $B(t+1) = (1 + r) \times B(t) - 0.03 \times B(t)$ Simplifying: $B(t+1) = (1 + 0.01833 - 0.03) \times B(t)$ $B(t+1) = 0.98833 \times B(t)$

4. General Formula for $B(t)$:

   • Since the balance is multiplied by $0.98833$ each month, the formula for the balance after $t$ months is an exponential decay: $B(t) = 5000 \times (0.98833)^t$

Final Formula:

The formula for the balance after $t$ months is: $B(t) = 5000 \times (0.98833)^t$

This formula can be used to calculate the remaining balance after any number of months, $t$.

Would you like more details on how this was derived? Here are some related questions for further exploration:

1. How can the formula be adjusted if the APR or minimum payment percentage changes?
2. What would be the balance after 12 months (1 year) using the current formula?
3. How long will it take for the balance to be reduced to half its initial value?
4. What happens if a higher payment (e.g., 5% of the balance) is made each month?
5. How does the effective interest rate change when considering daily compounding?

Tip: When calculating credit card balances, it's essential to understand the impact of interest compounding and minimum payments to manage debt effectively.