Use the compound interest formulas A = P (1+ r/n)^nt and A =Pe^rt to solve exercises 53-56. Round answers to the nearest cent. Find the accumulated value of an investment of $10,000 for 5 years at an interest rate of 1.32% if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.

Verified step by step guidance

1. Identify the given values: Principal \( P = 10,000 \), interest rate \( r = 0.0132 \), and time \( t = 5 \) years.
2. For part (a), use the formula for compound interest: \( A = P \left(1 + \frac{r}{n}\right)^{nt} \) with \( n = 2 \) for semiannual compounding.
3. For part (b), use the same formula with \( n = 4 \) for quarterly compounding.
4. For part (c), use the same formula with \( n = 12 \) for monthly compounding.
5. For part (d), use the continuous compounding formula: \( A = Pe^{rt} \).

Key Concepts

Compound Interest

Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. It differs from simple interest, which is calculated only on the principal amount. The frequency of compounding (e.g., annually, semiannually, quarterly, monthly) affects the total amount of interest earned or paid over time.

Compounding Frequency

Compounding frequency refers to how often interest is calculated and added to the principal balance of an investment or loan. Common compounding frequencies include annually, semiannually, quarterly, and monthly. The more frequently interest is compounded, the more interest will accumulate, leading to a higher total amount at the end of the investment period.

Continuous Compounding

Continuous compounding is a mathematical concept where interest is calculated and added to the principal at every possible instant, rather than at discrete intervals. The formula A = Pe^rt is used for continuous compounding, where 'e' is the base of the natural logarithm. This method results in the highest possible amount of interest earned compared to other compounding methods.