How much will deposits of ​$330 made at the end of each 6 months amount to after 6 years if interest is 8​% compounded annually​?
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Part 1
The deposits will amount to ​$

enter your response here.
​(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as​ needed.)

To solve this problem, we need to calculate the future value of an annuity where deposits are made semiannually. The formula to calculate the future value of an annuity with periodic deposits is:

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

Where:
- $ P $ is the amount of each deposit.
- $ r $ is the interest rate per period.
- $ n $ is the total number of periods.

### Step 1: Identify the values
- $ P = 330 $ (amount of each deposit)
- Annual interest rate $ = 8\% = 0.08 $
- Compounding frequency = 2 times per year (since the deposits are semiannual)
- Number of years $ = 6 $

### Step 2: Calculate the interest rate per period and the total number of periods
- $ r = \frac{0.08}{2} = 0.04 $ (since the interest is compounded annually but deposits are made every 6 months)
- Total number of periods $ n = 6 \times 2 = 12 $

### Step 3: Plug the values into the formula
\[
FV = 330 \times \left(\frac{(1 + 0.04)^{12} - 1}{0.04}\right)
\]

### Step 4: Calculate the future value
Let's calculate this step by step:

1. $ (1 + 0.04)^{12} = 1.601032 $ (rounded to six decimal places)
2. $ (1.601032 - 1) = 0.601032 $
3. $ \frac{0.601032}{0.04} = 15.0258 $
4. Finally, $ 330 \times 15.0258 = 4958.514 $

### Step 5: Round to the nearest cent
The final amount is approximately $ \text{\$4958.51} $.

So, the deposits will amount to **\$4,958.51** after 6 years.

Would you like any further details or have any questions?

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Here are some related questions:
1. How would the amount change if the interest was compounded monthly instead of annually?
2. What if the deposits were made quarterly instead of semiannually?
3. How does changing the interest rate to 10% affect the future value?
4. What if the time period was extended to 10 years instead of 6 years?
5. How much would the future value be if the deposit amount was $500 instead of $330?
6. What would be the future value if no interest was applied at all?
7. How does the effective annual rate (EAR) compare to the nominal rate in this scenario?
8. How would the calculations change if the interest was compounded continuously?

**Tip:** Always break down complex interest problems into smaller, manageable steps to avoid confusion.

Learn how to calculate the future value of semiannual deposits of $330 over 6 years with an 8% annual interest rate. Follow a step-by-step solution using the Future Value of an Annuity formula. Suitable for advanced high school students and learners interested in financial mathematics.

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