To what amount will $P invested for n years at an annual interest rate of r percent, compounded annually, accumulate?

• A. \(P(1 + r)^n\)
• B. \(P(1 + n)^r\)
• C. \(P \times r \times n\)
• D. \(P(1 + r \, n)\)

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the concept of compound interest. Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. The formula for compound interest is: \( A = P(1 + r)^n \), where \( A \) is the accumulated amount, \( P \) is the principal, \( r \) is the annual interest rate (in decimal form), and \( n \) is the number of years.
2. Step 2: Analyze the given options. The correct formula for compound interest is \( P(1 + r)^n \). The other options provided do not correctly represent the compounding process. For example, \( P(1 + n)^r \) incorrectly applies the interest rate as an exponent to the time period, and \( P \times r \times n \) represents simple interest, not compound interest.
3. Step 3: Break down the correct formula \( P(1 + r)^n \). Here, \( (1 + r) \) represents the growth factor for one year, and raising it to the power of \( n \) accounts for compounding over \( n \) years.
4. Step 4: Convert the annual interest rate \( r \) from a percentage to a decimal by dividing it by 100. For example, if \( r \) is 5%, then \( r = 0.05 \). This step ensures the formula is applied correctly.
5. Step 5: Substitute the values of \( P \), \( r \), and \( n \) into the formula \( A = P(1 + r)^n \) to calculate the accumulated amount. Ensure all values are correctly placed and the exponentiation is performed accurately.