interest earned on $ 20 comma 000
 
invested for 5
 
years at 6
​% 
interest compounded as follows.
a. Annually 
b. 
Semiannually​ (twice a​ year) 
c. Quarterly 
d. Monthly
e. Continuously

To calculate the interest earned on an investment of $20,000 over 5 years at an interest rate of 6%, compounded in different ways, we'll use the following formulas:

### Compound Interest Formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $A$ is the final amount (principal + interest).
- $P$ is the principal amount (initial investment).
- $r$ is the annual interest rate (decimal form).
- $n$ is the number of times the interest is compounded per year.
- $t$ is the time the money is invested for (in years).

### Continuously Compounded Interest Formula:
\[
A = P e^{rt}
\]
Where:
- $e$ is Euler's number (approximately 2.71828).

Now, let's calculate the interest for each case:

Given:
- $P = 20,000$
- $r = 6\% = 0.06$
- $t = 5$ years

### a. Annually Compounded:
For annual compounding, $n = 1$:
\[
A = 20,000 \left(1 + \frac{0.06}{1}\right)^{1 \times 5}
\]
\[
A = 20,000 (1.06)^5
\]

### b. Semiannually Compounded:
For semiannual compounding, $n = 2$:
\[
A = 20,000 \left(1 + \frac{0.06}{2}\right)^{2 \times 5}
\]
\[
A = 20,000 \left(1 + 0.03\right)^{10}
\]

### c. Quarterly Compounded:
For quarterly compounding, $n = 4$:
\[
A = 20,000 \left(1 + \frac{0.06}{4}\right)^{4 \times 5}
\]
\[
A = 20,000 \left(1 + 0.015\right)^{20}
\]

### d. Monthly Compounded:
For monthly compounding, $n = 12$:
\[
A = 20,000 \left(1 + \frac{0.06}{12}\right)^{12 \times 5}
\]
\[
A = 20,000 \left(1 + 0.005\right)^{60}
\]

### e. Continuously Compounded:
For continuous compounding:
\[
A = 20,000 e^{0.06 \times 5}
\]

I’ll now calculate the interest for each case.Here are the interest amounts earned on an investment of $20,000 over 5 years at an interest rate of 6%, based on different compounding methods:

a. **Annually Compounded**: $6,764.51  
b. **Semiannually Compounded**: $6,878.33  
c. **Quarterly Compounded**: $6,937.10  
d. **Monthly Compounded**: $6,977.00  
e. **Continuously Compounded**: $6,997.18

These results show how increasing the frequency of compounding slightly increases the total interest earned.

Would you like further details or explanations? 

Here are five related questions to deepen understanding:
1. How does the interest amount change with different compounding frequencies?
2. What is the impact of time on compound interest?
3. How would the interest differ if the rate were 8% instead of 6%?
4. What would the interest be if the investment duration was 10 years instead of 5?
5. Can the same formulas be applied to other financial scenarios?

**Tip**: Compounding more frequently results in slightly higher interest, but the difference diminishes as frequency increases.

Interest earned on $20,000 invested for 5 years at 6% interest compounded as follows: a. Annually b. Semiannually c. Quarterly d. Monthly e. Continuously

Calculate the interest earned on an investment of $20,000 at 6% interest over 5 years, compounded annually, semiannually, quarterly, monthly, and continuously. This example explains compound interest, exponential growth, and includes detailed calculations for each compounding frequency.

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