To calculate how much $14,500 will grow over 20 years with different compounding frequencies, we will use the compound interest formula:

$A = P \left( 1 + \frac{r}{n} \right)^{nt}$

Where:

• $A$ is the amount of money accumulated after interest.
• $P$ is the principal amount (the initial deposit).
• $r$ is the annual interest rate (decimal).
• $n$ is the number of times the interest is compounded per year.
• $t$ is the number of years the money is invested.

Given:

• $P = 14,500$
• $r = 0.04$
• $t = 20$ years

We'll calculate for each compounding frequency:

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a. 4% compounded annually (n = 1)

For annual compounding, $n = 1$.

$A = 14,500 \left( 1 + \frac{0.04}{1} \right)^{1 \times 20}$ $A = 14,500 \left( 1 + 0.04 \right)^{20}$ $A = 14,500 \left( 1.04 \right)^{20}$ $A = 14,500 \times 2.191123$ $A = 31,850.28$

So, the amount after 20 years is $31,850.28.

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b. 4% compounded semiannually (n = 2)

For semiannual compounding, $n = 2$.

$A = 14,500 \left( 1 + \frac{0.04}{2} \right)^{2 \times 20}$ $A = 14,500 \left( 1 + 0.02 \right)^{40}$ $A = 14,500 \left( 1.02 \right)^{40}$ $A = 14,500 \times 2.208040$ $A = 31,989.58$

So, the amount after 20 years is $31,989.58.

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c. 4% compounded quarterly (n = 4)

For quarterly compounding, $n = 4$.

$A = 14,500 \left( 1 + \frac{0.04}{4} \right)^{4 \times 20}$ $A = 14,500 \left( 1 + 0.01 \right)^{80}$ $A = 14,500 \left( 1.01 \right)^{80}$ $A = 14,500 \times 2.208040$ $A = 31,989.58$

So, the amount after 20 years is $31,989.58.

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d. 4% compounded monthly (n = 12)

For monthly compounding, $n = 12$.

$A = 14,500 \left( 1 + \frac{0.04}{12} \right)^{12 \times 20}$ $A = 14,500 \left( 1 + 0.003333 \right)^{240}$ $A = 14,500 \left( 1.003333 \right)^{240}$ $A = 14,500 \times 2.219640$ $A = 32,129.78$

So, the amount after 20 years is $32,129.78.

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Summary of Results:

a. 4% compounded annually: $31,850.28
b. 4% compounded semiannually: $31,989.58
c. 4% compounded quarterly: $31,989.58
d. 4% compounded monthly: $32,129.78

Would you like any further details or clarification?
Here are 5 related questions to expand on this:

1. How does the frequency of compounding affect the final amount?
2. What would happen to the amount if the interest rate were higher?
3. How would the amount change if the principal was increased or decreased?
4. How can you calculate the time needed to double your money with a given interest rate?
5. What are the practical advantages of monthly versus annual compounding?

Tip: Compounding more frequently generally leads to a higher accumulated value, even if the interest rate is the same.