Find the accumulated amount after 5 years if $2000 is invested at 3%/year compounded in the following ways. (Round your answers to the nearest cent.)

(a) compounded annually
$  

(b) compounded semiannually
$  

(c) compounded quarterly
$  

(d) compounded monthly
$  

To solve this problem, we can use the formula for compound interest:

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

Where:
- $ A $ = the accumulated amount (future value)
- $ P $ = the principal amount (initial investment), which is \$2000
- $ r $ = the annual interest rate (as a decimal), which is 3% or 0.03
- $ t $ = the time in years, which is 5 years
- $ n $ = the number of times the interest is compounded per year (varies for each part of the question)

### (a) Compounded annually

For annually compounded interest, $ n = 1 $.

\[
A = 2000 \left(1 + \frac{0.03}{1}\right)^{1 \times 5} = 2000 \times \left(1.03\right)^5
\]

\[
A \approx 2000 \times 1.159274 = 2318.55
\]

So, the accumulated amount is **\$2318.55**.

### (b) Compounded semiannually

For semiannually compounded interest, $ n = 2 $.

\[
A = 2000 \left(1 + \frac{0.03}{2}\right)^{2 \times 5} = 2000 \times \left(1.015\right)^{10}
\]

\[
A \approx 2000 \times 1.160968 = 2321.94
\]

So, the accumulated amount is **\$2321.94**.

### (c) Compounded quarterly

For quarterly compounded interest, $ n = 4 $.

\[
A = 2000 \left(1 + \frac{0.03}{4}\right)^{4 \times 5} = 2000 \times \left(1.0075\right)^{20}
\]

\[
A \approx 2000 \times 1.161976 = 2323.95
\]

So, the accumulated amount is **\$2323.95**.

### (d) Compounded monthly

For monthly compounded interest, $ n = 12 $.

\[
A = 2000 \left(1 + \frac{0.03}{12}\right)^{12 \times 5} = 2000 \times \left(1.0025\right)^{60}
\]

\[
A \approx 2000 \times 1.162889 = 2325.78
\]

So, the accumulated amount is **\$2325.78**.

### Summary of Results:
- (a) Compounded annually: **\$2318.55**
- (b) Compounded semiannually: **\$2321.94**
- (c) Compounded quarterly: **\$2323.95**
- (d) Compounded monthly: **\$2325.78**

Let me know if you want more details or have any questions!

Here are 5 related questions:
1. How does the number of compounding periods affect the total amount?
2. What is the formula for continuous compounding?
3. What happens to the accumulated amount if the interest rate is increased to 5%?
4. How would the accumulated amount change if the investment period was 10 years instead of 5?
5. What would happen if interest was compounded daily?

**Tip**: More frequent compounding leads to slightly higher accumulated amounts, though the differences diminish as the frequency increases.

Find the accumulated amount after 5 years if $2000 is invested at 3%/year compounded in the following ways. (Round your answers to the nearest cent.) (a) compounded annually (b) compounded semiannually (c) compounded quarterly (d) compounded monthly

Learn how to calculate the accumulated amount after 5 years for a $2000 investment at 3% annual interest rate, compounded annually, semiannually, quarterly, and monthly. This compound interest problem covers exponential growth and the key compound interest formula. Suitable for Grades 9-12.

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