Find the accumulated value of an investment of $ 20 comma 000 for 4 years at an interest rate of 1.55 % if the money is a. compounded​ semiannually; b. compounded​ quarterly; c. compounded monthly d. compounded continuously.

To calculate the accumulated value of an investment with compound interest, we can use the following formulas:

1. **For compound interest (discrete compounding)**:
   \[
   A = P \left(1 + \frac{r}{n}\right)^{nt}
   \]
   Where:
   - $A$ is the accumulated amount (future value).
   - $P$ is the principal amount (initial investment).
   - $r$ is the annual interest rate (as a decimal).
   - $n$ is the number of times the interest is compounded per year.
   - $t$ is the time the money is invested for, in years.

2. **For continuous compounding**:
   \[
   A = P e^{rt}
   \]
   Where:
   - $e$ is the mathematical constant (approximately 2.71828).

Now, let's calculate the accumulated value for each case with the given information:
- Principal ($P$) = 20,000
- Annual interest rate ($r$) = 1.55% = 0.0155
- Time ($t$) = 4 years

### a. Compounded Semiannually
Here, $n = 2$.
\[
A = 20,000 \left(1 + \frac{0.0155}{2}\right)^{2 \times 4}
\]

### b. Compounded Quarterly
Here, $n = 4$.
\[
A = 20,000 \left(1 + \frac{0.0155}{4}\right)^{4 \times 4}
\]

### c. Compounded Monthly
Here, $n = 12$.
\[
A = 20,000 \left(1 + \frac{0.0155}{12}\right)^{12 \times 4}
\]

### d. Compounded Continuously
For continuous compounding:
\[
A = 20,000 \times e^{0.0155 \times 4}
\]

Let me calculate each value for you.The accumulated value of the investment of $20,000 for 4 years at an interest rate of 1.55% is:

a. **Compounded semiannually**: \$21,274.16  
b. **Compounded quarterly**: \$21,276.70  
c. **Compounded monthly**: \$21,278.40  
d. **Compounded continuously**: \$21,279.25

Would you like more details on any of these calculations or have further questions?

Here are 5 related questions to expand on the topic:
1. How would the accumulated value change if the interest rate increased to 2%?
2. What happens to the accumulated value if the investment duration is 10 years instead of 4?
3. How does compounding daily affect the total accumulated value compared to monthly compounding?
4. Can you calculate the accumulated value for a different principal, say $50,000, for the same 4-year period?
5. How do taxes affect the final value of the investment in this scenario?

**Tip**: The more frequently interest is compounded, the higher the accumulated value. Continuous compounding results in the maximum possible return.

Find the accumulated value of an investment of $20,000 for 4 years at an interest rate of 1.55% if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.

Learn how to calculate the accumulated value of an investment of $20,000 over 4 years with a 1.55% interest rate. This problem covers compound interest (semiannually, quarterly, monthly) and continuous compounding using the relevant formulas. Suitable for students in Grades 10-12.

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