Final answer:
To determine the monthly payments the student will receive for 10 years after a $81,000 investment at 12% annual interest rate compounded weekly, we use the compound interest formula to calculate the amount after 7 years, which then serves as the present value in the annuity formula to calculate the monthly payments.
Step-by-step explanation:
To solve this problem, we must first understand compound interest and how it applies to the situation described. Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. In this case, the interest is compounded weekly, which impacts the final amount differently than if it were compounded annually.
The student's initial investment is $81,000 at a 12% annual interest rate, compounded weekly. After 7 years, the future value of this investment can be calculated using the compound interest formula:
FV = P × (1 + r/n)^(nt)
Where:
The future value after 7 years is now effectively the present value for the annuity calculation, as the student wishes to receive equal monthly payments for the next 10 years. An annuity formula can be applied to determine the monthly payments:
PMT = (PV × (r/n)) / (1 - (1 + r/n)^(-nt))
Where:
Using this annuity formula and the future value obtained earlier, we can calculate the monthly payment that the student will receive for the next 10 years.
While the specific numerical answer will require calculation, it's important to remember that compound interest can make a huge difference with larger sums of money and over longer periods, as its effects are more significant than those of simple interest.