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. \(P(1 + r)^n\)
• C. \(P(1 - r)^n\)
• D. \(P \times r \times n\)

Answer: (B)

Verified step by step guidance

1. Understand the problem: The question is asking for the formula to calculate the future value of an investment ($P) when it is compounded annually at an annual interest rate (r) for n years.
2. Recall the formula for compound interest: The future value (FV) of an investment compounded annually is given by the formula: $FV\; =\; P(1\; +\; r)^n$, where P is the principal amount, r is the annual interest rate (expressed as a decimal), and n is the number of years.
3. Analyze the options provided: The correct formula must include the compounding effect, which is represented by raising (1 + r) to the power of n. This accounts for the interest being applied to the principal and previously earned interest each year.
4. Eliminate incorrect options: The formula $P(1\; +\; r\; \backslash ,\; n)$ is incorrect because it implies simple interest, not compound interest. Similarly, $P\; \backslash times\; r\; \backslash times\; n$ represents simple interest, not compounding. The formula $P(1\; -\; r)^n$ is incorrect because it suggests a decrease in value, which is not relevant here.
5. Conclude with the correct formula: The correct formula is $P(1\; +\; r)^n$, as it properly accounts for the compounding of interest annually over n years.