https://file.aiquickdraw.com/mathbot/file/1727382142185sc23cuiz.jpg

Let's solve the questions provided in the image one by one:

### **Part A**
We are given:
- Initial investment ($P$): \$10,000
- Annual interest rate ($r$): 4.00% (0.04 as a decimal)
- Compounded semi-annually: meaning there are 2 compounding periods per year ($n = 2$)
- Time ($t$): 5 years

The formula to calculate the future value with compound interest is:

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

Substituting the values:

\[
A = 10,000 \left(1 + \frac{0.04}{2}\right)^{2 \times 5}
\]

Let me calculate this value for you.

### **Part B**
In this part, the interest rate increases by 2% per year, so the interest rate for year 1 is 4.00%, year 2 is 4.08%, and so on.

We will need to calculate the future value year by year, updating the interest rate at the end of each year. Let me compute this step by step.

I will start by calculating both parts for you now.### Results:

**Part A:**  
The future value of your investment after 5 years with a 4% interest rate compounded semi-annually is approximately **\$12,189.94**.

**Part B:**  
Assuming the interest rate increases by 2% per year, the future value of your investment after 10 years is approximately **\$15,422.57**.

Would you like further details or explanations on any part of the calculations?

Here are 5 related questions you may find useful:
1. What would happen if the interest were compounded quarterly instead of semi-annually?
2. How does continuous compounding affect the future value?
3. What is the impact of a higher initial investment, say \$20,000, on the future values?
4. How does changing the number of compounding periods per year affect the future value in Part A?
5. What would the future value be if the interest rate increased by 1% per year instead of 2%?

**Tip:**  
When interest rates are compounded more frequently, your investment grows faster due to the increased number of compounding periods.

The interest rate quoted by a bank is 4.00% APR, compounded semi-annually. You decide to invest $10,000 with this bank. Part A asks for the future value of the investment in 5 years at a constant interest rate. Part B asks for the future value of the investment in 10 years, assuming the interest rate increases by 2% annually.

Learn how to calculate the future value of a $10,000 investment with a 4.00% APR compounded semi-annually. Explore Part A for a constant interest rate over 5 years and Part B where the APR increases by 2% per year for 10 years. Ideal for high school and college students.

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