Use the compound interest formulas 𝐴=𝑃(1+𝑟𝑛)𝑛𝑡 and 𝐴=𝑃𝑒𝑟𝑡 to solve the exercises below.
Round your answer to the nearest cent.
Find the accumulated value of an investment of $14490 for 7 years at an annual interest rate of 7.1% if the money is
(a) compounded semiannually: dollars
(b) compounded quarterly: dollars
(c) compounded monthly: dollars
(d) compounded continuously:  dollars

To solve this, we will use the compound interest formulas:

1. **For compound interest formula (non-continuous):**  
   \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

   Where:  
   - $ A $ is the accumulated amount (future value).  
   - $ P $ is the principal amount (initial investment).  
   - $ r $ is the annual interest rate (expressed as a decimal).  
   - $ n $ is the number of times the interest is compounded per year.  
   - $ t $ is the number of years.  

2. **For continuously compounded interest formula:**  
   \[ A = P \cdot e^{rt} \]

   Where:  
   - $ A $ is the accumulated amount.  
   - $ P $ is the principal amount.  
   - $ r $ is the annual interest rate (expressed as a decimal).  
   - $ t $ is the number of years.  
   - $ e $ is Euler's number, approximately 2.71828.  

### Given Information:
- Principal ($ P $): \$14,490
- Annual interest rate ($ r $): 7.1% or 0.071 (in decimal)
- Time ($ t $): 7 years

Now, let's solve each part:

### (a) Compounded Semiannually:
- $ n = 2 $ (compounded twice a year)

\[
A = 14490 \left(1 + \frac{0.071}{2}\right)^{2 \cdot 7}
\]
\[
A = 14490 \left(1 + 0.0355\right)^{14}
\]
\[
A = 14490 \times (1.0355)^{14}
\]
\[
A \approx 14490 \times 1.322994
\]
\[
A \approx 19165.94
\]

### (b) Compounded Quarterly:
- $ n = 4 $ (compounded four times a year)

\[
A = 14490 \left(1 + \frac{0.071}{4}\right)^{4 \cdot 7}
\]
\[
A = 14490 \left(1 + 0.01775\right)^{28}
\]
\[
A = 14490 \times (1.01775)^{28}
\]
\[
A \approx 14490 \times 1.327568
\]
\[
A \approx 19227.51
\]

### (c) Compounded Monthly:
- $ n = 12 $ (compounded twelve times a year)

\[
A = 14490 \left(1 + \frac{0.071}{12}\right)^{12 \cdot 7}
\]
\[
A = 14490 \left(1 + 0.00591667\right)^{84}
\]
\[
A = 14490 \times (1.00591667)^{84}
\]
\[
A \approx 14490 \times 1.329926
\]
\[
A \approx 19262.95
\]

### (d) Compounded Continuously:
\[
A = 14490 \cdot e^{0.071 \cdot 7}
\]
\[
A = 14490 \cdot e^{0.497}
\]
\[
A \approx 14490 \cdot 1.6445
\]
\[
A \approx 19290.11
\]

### Summary:
- **(a) Compounded semiannually:** \$19,165.94
- **(b) Compounded quarterly:** \$19,227.51
- **(c) Compounded monthly:** \$19,262.95
- **(d) Compounded continuously:** \$19,290.11

Would you like more details on any step or have further questions? Here are five related questions you might find interesting:

1. How does increasing the number of compounding periods per year affect the accumulated value?
2. What happens if the interest rate is compounded weekly instead of semiannually or quarterly?
3. What formula would you use for a different investment period, say 10 years?
4. How does the interest rate impact the accumulated amount if compounded at different intervals?
5. What practical scenarios use continuous compounding instead of periodic compounding?

**Tip:** Compounding frequency has a significant impact on your accumulated amount; the more frequently interest is compounded, the higher your total will be!

Find the accumulated value of an investment of $14490 for 7 years at an annual interest rate of 7.1% if the money is (a) compounded semiannually (b) compounded quarterly (c) compounded monthly (d) compounded continuously.

Learn how to calculate the accumulated value of a $14,490 investment over 7 years at an interest rate of 7.1%. Explore the differences between compounding semiannually, quarterly, monthly, and continuously using compound interest formulas.

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