Find the FV of $1,000 invested to earn 10% after 4 years with semiannual compounding. Round your answer to the nearest cent.

To calculate the future value (FV) of an investment using semiannual compounding, we can use the formula:

\[
FV = P \times \left(1 + \frac{r}{n}\right)^{n \times t}
\]

Where:
- $P$ = the principal amount (initial investment) = $1,000
- $r$ = annual interest rate = 10% or 0.10
- $n$ = number of compounding periods per year = 2 (since it's semiannual compounding)
- $t$ = number of years = 4

Now, we can plug in the values:

\[
FV = 1000 \times \left(1 + \frac{0.10}{2}\right)^{2 \times 4}
\]

Let's calculate this.The future value (FV) of the $1,000 investment, compounded semiannually at a 10% annual interest rate after 4 years, is approximately **$1,477.46**.

Would you like a more detailed explanation, or do you have any further questions?

Here are five related questions for further exploration:
1. What would be the future value with quarterly compounding instead?
2. How does the frequency of compounding affect the future value?
3. How much interest would be earned on this investment over the 4 years?
4. What would be the future value if the interest rate were 12% instead of 10%?
5. How does continuous compounding compare to semiannual compounding in this scenario?

**Tip:** The more frequent the compounding, the higher the future value will be, as interest is being calculated on accumulated interest more often.

Calculate the future value of $1,000 invested at 10% annual interest with semiannual compounding over 4 years. This problem uses the compound interest formula to show how compounding frequency impacts the final amount. Suitable for students in Grades 10-12 or anyone learning about financial mathematics.

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