Which of the following equations correctly calculates the future value (\(FV\)) of a single sum invested today (\(PV\)) for \(n\) periods at an annual interest rate \(r\), with interest compounded once per period?

• A. \(FV = PV \times (1 - r)^n\)
• B. \(FV = PV \div (1 + r)^n\)
• C. \(FV = PV \times r \times n\)
• D. \(FV = PV \times (1 + r)^n\)

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the concept of future value (FV). Future value represents the amount of money an investment will grow to over time, given a specific interest rate and compounding period.
2. Step 2: Recognize the formula for future value with compound interest. The correct formula is \(FV = PV \times (1 + r)^n\), where \(PV\) is the present value, \(r\) is the annual interest rate, and \(n\) is the number of compounding periods.
3. Step 3: Analyze why the other formulas are incorrect. For example, \(FV = PV \times (1 - r)^n\) incorrectly subtracts the interest rate, which does not reflect growth. Similarly, \(FV = PV \div (1 + r)^n\) represents a discounting formula, not compounding. Lastly, \(FV = PV \times r \times n\) assumes simple interest, not compound interest.
4. Step 4: Break down the correct formula. The term \((1 + r)^n\) accounts for compounding, where the investment grows exponentially over \(n\) periods. Multiplying this by \(PV\) scales the growth to the initial investment amount.
5. Step 5: Apply the formula to solve problems. To calculate future value, substitute the given values for \(PV\), \(r\), and \(n\) into the formula \(FV = PV \times (1 + r)^n\) and compute the result.