To solve these problems, we will use the following formulas for each compounding option:

1. Compounded Annually, Monthly, and Daily:
   The formula is: $A = P \left(1 + \frac{r}{n}\right)^{nt}$ where:

   • $A$ is the amount after time $t$,
   • $P$ is the principal (initial amount, here $10,000),
   • $r$ is the annual interest rate (5.2% or 0.052),
   • $n$ is the number of times the interest is compounded per year (1 for annually, 12 for monthly, 365 for daily),
   • $t$ is the number of years (5 years in this case).

2. Compounded Continuously: The formula is: $A = P \cdot e^{rt}$ where $e$ is the base of the natural logarithm (approximately 2.71828).

Let’s go through each option:

a. Compounded Annually ($n = 1$)

$A = 10000 \left(1 + \frac{0.052}{1}\right)^{1 \times 5}$ $A = 10000 \times (1.052)^5$ Calculating this gives: $A \approx 10000 \times 1.288268$ $A \approx 12882.68$

b. Compounded Monthly ($n = 12$)

$A = 10000 \left(1 + \frac{0.052}{12}\right)^{12 \times 5}$ $A = 10000 \times \left(1 + 0.0043333\right)^{60}$ $A \approx 10000 \times 1.289961$ $A \approx 12899.61$

c. Compounded Daily ($n = 365$)

$A = 10000 \left(1 + \frac{0.052}{365}\right)^{365 \times 5}$ $A = 10000 \times \left(1 + 0.0001424658\right)^{1825}$ $A \approx 10000 \times 1.290429$ $A \approx 12904.29$

d. Compounded Continuously

$A = 10000 \cdot e^{0.052 \times 5}$ $A = 10000 \cdot e^{0.26}$ Calculating $e^{0.26} \approx 1.29693$: $A \approx 10000 \times 1.29693$ $A \approx 12969.30$

Summary

• Compounded Annually: $12,882.68
• Compounded Monthly: $12,899.61
• Compounded Daily: $12,904.29
• Compounded Continuously: $12,969.30

Would you like a breakdown of each calculation step?

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Related Questions:

1. What happens to the final amount if the interest rate increases to 6%?
2. How does changing the compounding period affect the growth rate?
3. What would the final amount be if invested for 10 years instead of 5?
4. How is continuous compounding derived mathematically from discrete compounding?
5. What if the interest rate were compounded quarterly instead of monthly?

Tip: Continuous compounding often results in the highest yield due to the nature of exponential growth with no interval between compounding