Find the interest earned on $ 20 comma 000 invested for 4 years at 5​% interest compounded as follows.
a. Annually 
b. Semiannually​ (twice a​ year) 
c. Quarterly 
d. Monthly
e. Continuously

To find the interest earned on an investment of $20,000 for 4 years at 5% interest compounded at different intervals, we will use the compound interest formula and the continuous compound interest formula:

### 1. **Compound Interest Formula:**

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

Where:
- $ A $ is the final amount (principal + interest),
- $ P $ is the principal (initial investment),
- $ r $ is the annual interest rate (as a decimal),
- $ n $ is the number of times interest is compounded per year,
- $ t $ is the number of years.

### 2. **Continuous Compound Interest Formula:**

\[
A = P e^{rt}
\]

Where $ e $ is Euler's number (approximately 2.71828).

Now, let's calculate each case:

### a. **Annually (n = 1)**

\[
A = 20,000 \left( 1 + \frac{0.05}{1} \right)^{1 \times 4}
\]
\[
A = 20,000 \left( 1.05 \right)^4
\]
\[
A \approx 20,000 \times 1.21550625
\]
\[
A \approx 24,310.13
\]

The interest earned is:
\[
24,310.13 - 20,000 = 4,310.13
\]

### b. **Semiannually (n = 2)**

\[
A = 20,000 \left( 1 + \frac{0.05}{2} \right)^{2 \times 4}
\]
\[
A = 20,000 \left( 1.025 \right)^8
\]
\[
A \approx 20,000 \times 1.218402
\]
\[
A \approx 24,368.04
\]

The interest earned is:
\[
24,368.04 - 20,000 = 4,368.04
\]

### c. **Quarterly (n = 4)**

\[
A = 20,000 \left( 1 + \frac{0.05}{4} \right)^{4 \times 4}
\]
\[
A = 20,000 \left( 1.0125 \right)^{16}
\]
\[
A \approx 20,000 \times 1.219391
\]
\[
A \approx 24,387.82
\]

The interest earned is:
\[
24,387.82 - 20,000 = 4,387.82
\]

### d. **Monthly (n = 12)**

\[
A = 20,000 \left( 1 + \frac{0.05}{12} \right)^{12 \times 4}
\]
\[
A = 20,000 \left( 1.0041667 \right)^{48}
\]
\[
A \approx 20,000 \times 1.220388
\]
\[
A \approx 24,407.76
\]

The interest earned is:
\[
24,407.76 - 20,000 = 4,407.76
\]

### e. **Continuously**

Using the continuous compound interest formula:

\[
A = 20,000 e^{0.05 \times 4}
\]
\[
A = 20,000 e^{0.20}
\]
\[
A \approx 20,000 \times 1.221403
\]
\[
A \approx 24,428.06
\]

The interest earned is:
\[
24,428.06 - 20,000 = 4,428.06
\]

### Summary of Interest Earned:
- **Annually:** $4,310.13
- **Semiannually:** $4,368.04
- **Quarterly:** $4,387.82
- **Monthly:** $4,407.76
- **Continuously:** $4,428.06

Would you like more details or have any questions?

### 5 Related Questions:
1. How would the interest change if the investment period was 5 years instead of 4?
2. What happens to the interest if the rate increases to 6% compounded quarterly?
3. How does monthly compounding compare to daily compounding for the same period?
4. What is the effective annual rate (EAR) for each compounding frequency?
5. How is continuous compounding used in real-world financial applications?

### Tip:
The more frequent the compounding period, the greater the amount of interest earned. Continuous compounding yields the highest interest.

Find the interest earned on $20,000 invested for 4 years at 5% interest compounded as follows:
a. Annually
b. Semiannually (twice a year)
c. Quarterly
d. Monthly
e. Continuously

Calculate the interest earned on $20,000 invested for 4 years at 5% interest, compounded annually, semiannually, quarterly, monthly, and continuously. Understand the key differences in interest amounts using the compound interest formula and the continuous compound interest formula. Ideal for high school students learning about exponential growth.

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