We can solve this problem using the formula for compound interest:

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

where:

• $A$ is the final amount after interest,
• $P$ is the initial principal (the amount deposited),
• $r$ is the annual interest rate (expressed as a decimal),
• $n$ is the number of times the interest is compounded per year,
• $t$ is the number of years.

In this case:

• $P = 14,000$,
• $r = 0.03$ (since 3% APR),
• $n = 4$ (quarterly compounding),
• $t = 6$ years.

Step 1: Plug the values into the formula.

$A = 14,000 \left(1 + \frac{0.03}{4}\right)^{4 \times 6}$ $A = 14,000 \left(1 + 0.0075\right)^{24}$ $A = 14,000 \left(1.0075\right)^{24}$

Step 2: Calculate the exponent.

First, calculate $(1.0075)^{24}$.

$(1.0075)^{24} \approx 1.196682$

Step 3: Calculate the final amount.

$A \approx 14,000 \times 1.196682 = 16,753.55$

Step 4: Find the interest earned.

The interest earned is the difference between the final amount and the initial principal:

$\text{Interest} = A - P = 16,753.55 - 14,000 = 2,753.55$

Final Answer:

The interest earned is approximately $2,753.55.

Would you like further details on any step, or do you have questions?

---

Here are 5 related questions for practice:

1. What is the formula for continuously compounded interest?
2. How does increasing the number of compounding periods affect the final amount?
3. What would the interest be if the APR were 5% instead of 3%?
4. How would the result change if the interest were compounded annually instead of quarterly?
5. What is the relationship between APR and APY (Annual Percentage Yield)?

Tip: Compounding more frequently increases the amount of interest earned, as each interest calculation includes previously earned interest.