You have just graduated from college and landed your first big job. You have always dreamed of being a homeowner, and after carefully shopping for your dream home, you find one that you would like to purchase at a cost of $250,000. After researching banks to find the best interest rate, you find that Banks for Homeowners offers the best rate of 6% interest that compounds monthly for 30 years.

* What is the monthly payment for this loan?
* What is the unpaid balance of the loan at the end of 5 years?
* What is the unpaid balance at the end of the 10th year?

To calculate the monthly payment for the loan, we can use the formula for the monthly payment of a fixed-rate mortgage:

M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1 ]

Where:
M = Monthly payment
P = Loan amount ($250,000)
i = Monthly interest rate (6% / 12 = 0.005)
n = Number of monthly payments (30 years * 12 months = 360)

Now we can substitute these values into the formula:

M = 250000 [ 0.005(1 + 0.005)^360 ] / [ (1 + 0.005)^360 - 1 ]

Calculating this expression will give us the monthly payment for the loan.

To calculate the unpaid balance of the loan at the end of 5 years, we can use the loan amortization formula:

B = P [ (1 + i)^n - (1 + i)^p ] / [ (1 + i)^n - 1 ]

Where:
B = Unpaid balance
P = Loan amount ($250,000)
i = Monthly interest rate (0.005)
n = Number of monthly payments (360)
p = Number of monthly payments made (5 years * 12 months = 60)

Substituting the values into the formula will give us the unpaid balance at the end of 5 years.

To calculate the unpaid balance at the end of the 10th year, we will follow a similar process using the loan amortization formula but with p = 10 years * 12 months = 120.

By plugging in the values into the formula, we can find the unpaid balance at the end of the 10th year.

Note: When making these calculations, it's important to include any additional fees or costs associated with the loan, as well as considering the possible impact of taxes, insurance, or other factors that may affect the final payments and unpaid balances.