Final answer:
To find the accumulated value of a $15,000 investment at 6.5% interest, apply the compound interest formula for each compounding frequency: semiannual, quarterly, monthly, and continuous. Compound interest can greatly enhance the growth of an investment over time, especially with a larger initial sum and a longer duration.
Step-by-step explanation:
The accumulated value of an investment depends on the interest rate and the frequency with which the interest is compounded. To find the accumulated value for different compound frequencies:
- Determine the number of compounding periods per year (n).
- Divide the annual interest rate by n to find the periodic interest rate (r).
- Calculate the total number of compounding periods by multiplying n by the number of years (t).
- Apply the compound interest formula Accumulated Value = P(1 + r/n)^(nt), where P is the principal amount.
Using this method:
- For semiannual compounding (n=2), the accumulated value would be $15,000(1 + 0.065/2)^(2*7).
- For quarterly compounding (n=4), the accumulated value would be $15,000(1 + 0.065/4)^(4*7).
- For monthly compounding (n=12), the accumulated value would be $15,000(1 + 0.065/12)^(12*7).
- For continuous compounding, the formula is A = Pe^(rt), where A is the Accumulated Value, P is the principal, e is Euler's number (approximately 2.71828), r is the annual interest rate, and t is the time in years. So, the accumulated value would be $15,000e^(0.065*7).
Remember that compound interest can significantly increase your investment especially over a long period of time and with a larger initial sum, as opposed to simple interest. Be sure to round your final values to the nearest tenth as per the question's instructions.