Final answer:
To determine the nominal interest rate compounded weekly, we calculate the future value of the first $100 deposit compounded monthly at 4% for 20 years, then determine how much the second $100 deposit contributed to the final amount of $500 over the last 10 years. Subtracting the first deposit's future value from $500 leaves the remainder from the second deposit, for which we use the future value formula with weekly compounding to solve for the unknown interest rate.
Step-by-step explanation:
Finding the Weekly Compounded Nominal Interest Rate
To solve this problem, we need to calculate the future value of two separate deposits under different interest compounding scenarios. The first deposit of $100 is compounded monthly at a nominal discount rate of 4% for 20 years. The second deposit of $100 is made at the end of these 20 years and the total amount is then compounded weekly for the remaining 10 years at an unknown nominal interest rate.
The future value of the first deposit after 20 years can be calculated using the monthly compounded interest formula:
FV = P × (1 + r/n)(nt), where P is the principal amount, r is the annual nominal interest rate, n is the number of times the interest is compounded per year, and t is the number of years. After calculating this, we subtract the result from the total final amount of $500 to find out how much the second deposit has yielded over the last 10 years with weekly compounding.
Finally, we use the future value formula for weekly compounding to solve for the unknown nominal interest rate. The formula is the same as before, but with n reflecting the number of weeks in a year.
If we denote the unknown weekly nominal interest rate as r_weekly, we can set up the equation that represents the state of the account after 30 years and solve for r_weekly.