To solve this, we will use the compound interest formula for parts (a), (b), and (c) and the continuous compound interest formula for part (d).

The two formulas are:

1. Compound Interest Formula (n times per year): $A = P \left( 1 + \frac{r}{n} \right)^{nt}$ where:

   • $A$: Accumulated amount
   • $P$: Principal amount ($10,000$ in this case)
   • $r$: Annual interest rate (4.5% = 0.045)
   • $n$: Number of compounding periods per year
   • $t$: Time in years (3 years)

2. Continuous Compound Interest Formula: $A = P e^{rt}$ where:

   • $e$: Euler's number ($\approx 2.718$)

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Part (a): Semiannual Compounding ($n = 2$)

$A = 10,000 \left( 1 + \frac{0.045}{2} \right)^{2 \cdot 3}$ $A = 10,000 \left( 1 + 0.0225 \right)^6$ $A = 10,000 \cdot (1.0225)^6$ $A \approx 10,000 \cdot 1.139693$ $A \approx 11,396.93$

The accumulated value is 11,396.93.

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Part (b): Quarterly Compounding ($n = 4$)

$A = 10,000 \left( 1 + \frac{0.045}{4} \right)^{4 \cdot 3}$ $A = 10,000 \left( 1 + 0.01125 \right)^{12}$ $A = 10,000 \cdot (1.01125)^{12}$ $A \approx 10,000 \cdot 1.143006$ $A \approx 11,430.06$

The accumulated value is 11,430.06.

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Part (c): Monthly Compounding ($n = 12$)

$A = 10,000 \left( 1 + \frac{0.045}{12} \right)^{12 \cdot 3}$ $A = 10,000 \left( 1 + 0.00375 \right)^{36}$ $A = 10,000 \cdot (1.00375)^{36}$ $A \approx 10,000 \cdot 1.144896$ $A \approx 11,448.96$

The accumulated value is 11,448.96.

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Part (d): Continuous Compounding

$A = 10,000 \cdot e^{0.045 \cdot 3}$ $A = 10,000 \cdot e^{0.135}$ Using $e^{0.135} \approx 1.144215$: $A \approx 10,000 \cdot 1.144215$ $A \approx 11,442.15$

The accumulated value is 11,442.15.

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Final Results:

1. Semiannual Compounding: $11,396.93$
2. Quarterly Compounding: $11,430.06$
3. Monthly Compounding: $11,448.96$
4. Continuous Compounding: $11,442.15$

If you need further clarifications or have additional questions, feel free to ask! 😊

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Follow-Up Questions:

1. How does the frequency of compounding affect the accumulated value?
2. Why is continuous compounding slightly less than monthly compounding in this case?
3. What would happen if the time $t$ increased from 3 years to 10 years?
4. What is the formula to find the effective annual rate (EAR) for compounded interest?
5. How does the principal amount $P$ affect the accumulated value?

Tip: Increasing the compounding frequency increases the accumulated value, but the impact diminishes as the frequency becomes very high.