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 $A=P{(1+\frac{r}{n})}^{nt}$ 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 $A=P{e}^{rt}$.

Key Concepts

Compound Interest

Compound interest refers to the interest calculated on the initial principal and also on the accumulated interest from previous periods. This means that interest is earned on both the original amount and the interest that has been added to it, leading to exponential growth over time. The frequency of compounding (e.g., annually, semiannually, quarterly, monthly) affects the total amount accumulated.

The Compound Interest Formula

The compound interest formula A = P (1 + r/n)^(nt) is used to calculate the accumulated amount A after t years, where P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. This formula allows for different compounding frequencies, which can significantly impact the final amount.

Continuous Compounding

Continuous compounding is a method where interest is calculated and added to the principal an infinite number of times per year. The formula A = Pe^(rt) is used for this calculation, where e is the base of the natural logarithm. This method results in the highest possible amount of interest accrued over time, as it assumes that interest is being compounded at every possible moment.