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The problem in the image asks to find the accumulated value of an investment of $20,000 for 6 years at an interest rate of 1.45%, using different compounding methods: semiannually, quarterly, monthly, and continuously.

### Formulas to Use:
1. **Compound Interest Formula (for non-continuous compounding):**
   \[
   A = P \left(1 + \frac{r}{n}\right)^{nt}
   \]
   - $A$ is the accumulated value.
   - $P$ is the principal (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 number of years.

2. **Continuous Compounding Formula:**
   \[
   A = P e^{rt}
   \]
   - $e$ is the base of the natural logarithm (~2.71828).
   - Other variables are the same as above.

### Given Information:
- Principal ($P$) = $20,000
- Rate ($r$) = 1.45% = 0.0145
- Time ($t$) = 6 years

#### a. **Compounded Semiannually** ($n = 2$): 
Use the compound interest formula.

#### b. **Compounded Quarterly** ($n = 4$): 
Same formula with $n = 4$.

#### c. **Compounded Monthly** ($n = 12$): 
Use $n = 12$.

#### d. **Compounded Continuously**: 
Use the continuous compounding formula.

Let me compute each part for you!Here are the accumulated values for each compounding method:

a. **Compounded Semiannually**: $21,811.09  
b. **Compounded Quarterly**: $21,814.50  
c. **Compounded Monthly**: $21,816.79  
d. **Compounded Continuously**: $21,817.93  

Would you like further details or explanations? Let me know!

Here are 5 related questions to deepen understanding:
1. What happens to the accumulated value if the interest rate increases?
2. How does decreasing the compounding frequency affect the accumulated value?
3. What is the significance of continuous compounding in real-life scenarios?
4. How would the accumulated value change if the investment period were shorter?
5. Can you calculate the accumulated value for daily compounding?

**Tip**: The more frequently interest is compounded, the higher the accumulated value, but the differences become smaller with higher frequencies.

Find the accumulated value of an investment of $20,000 for 6 years at an interest rate of 1.45% if the money is compounded semiannually, quarterly, monthly, and continuously.

Learn how to calculate the accumulated value of a $20,000 investment over 6 years at a 1.45% interest rate, compounded semiannually, quarterly, monthly, and continuously. Explore the compound interest and continuous compounding formulas, and understand how different compounding frequencies affect the final value.

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