Final answer:
To find the accumulated value of the annuity at the end of fifty years, we need to calculate the accumulated value of the increasing payments for the first 24 years and the level payments for the remaining 26 years. Using the compound interest formula, we find that the accumulated value of the increasing payments is $5,674.87, and the accumulated value of the level payments is $9,666.05. Adding these values together gives us a total accumulated value of $15,340.92.
Step-by-step explanation:
An annuity is a series of equal payments made at regular intervals. In this case, Serena receives a fifty-year annuity-due, which means the payments start at $2,000 and increase by 3% per year through the twenty-fourth payment. After the twenty-fourth payment, the payments stay at a level of $4,000.
To find the accumulated value of this annuity at the end of fifty years, we can break it down into two parts. First, we calculate the accumulated value of the increasing payments for the first 24 years using compound interest formula:
A = P(1 + r/n)^(nt), where A is the accumulated value, P is the initial payment, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.
For the increasing payments, P = $2,000, r = 4.2% = 0.042, n = 1, and t = 24. Plugging in these values, we get:
A = $2,000(1 + 0.042/1)^(1*24)
A = $5,674.87
For the level payments after the twenty-fourth payment, we can find the accumulated value using the same formula with P = $4,000, r = 4.2%, n = 1, and t = 26 (50 - 24).
Plugging in these values, we get:
A = $4,000(1 + 0.042/1)^(1*26)
A = $9,666.05
Finally, we add up the accumulated values of the increasing payments and the level payments:
Total accumulated value = $5,674.87 + $9,666.05 = $15,340.92