Find the accumulated value of an investment of $20,000 for 7 years at an interest rate of 1.65 % if the money is a. compounded​ semiannually; b. compounded​ quarterly; c. compounded monthly d. compounded continuously.
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Part 1
a. What is the accumulated value if the money is compounded​ semiannually?
​$22,438.06
​(Round to the nearest cent as​ needed.)
Part 2
b. What is the accumulated value if the money is compounded​ quarterly?

To find the accumulated value of an investment using compound interest, we can apply the formula:

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

Where:
- $A$ is the accumulated value (final amount),
- $P$ is the principal (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 (in years).

Given:
- $P = 20,000$,
- $r = 1.65\% = 0.0165$,
- $t = 7$ years.

### Part (a): Compounded Semiannually

For semiannual compounding, $n = 2$:

\[
A = 20,000 \left(1 + \frac{0.0165}{2}\right)^{2 \times 7}
\]

\[
A = 20,000 \left(1 + 0.00825\right)^{14}
\]

\[
A = 20,000 \left(1.00825\right)^{14}
\]

Using a calculator:

\[
A \approx 22,438.06
\]

(Which matches the given result.)

### Part (b): Compounded Quarterly

For quarterly compounding, $n = 4$:

\[
A = 20,000 \left(1 + \frac{0.0165}{4}\right)^{4 \times 7}
\]

\[
A = 20,000 \left(1 + 0.004125\right)^{28}
\]

\[
A = 20,000 \left(1.004125\right)^{28}
\]

Let's calculate this.The accumulated value when the money is compounded quarterly is approximately **$22,443.36**.

Would you like to proceed with part (c) for monthly compounding and part (d) for continuous compounding?

Here are some related questions:
1. How does changing the compounding frequency affect the accumulated value?
2. What is the difference between simple and compound interest?
3. How is continuous compounding different from discrete compounding methods?
4. What happens if the interest rate increases?
5. How does the time period affect the accumulated value in compound interest?

**Tip**: When comparing investments, the compounding frequency can significantly influence the total interest earned over time.

Find the accumulated value of an investment of $20,000 for 7 years at an interest rate of 1.65% if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.

Discover how to calculate the accumulated value of a $20,000 investment over 7 years at a 1.65% interest rate, compounded semiannually, quarterly, monthly, and continuously. This comprehensive guide covers essential formulas and provides step-by-step solutions suitable for high school students.

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