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Please give me the correct answer with accurate calculation.
You must round to the nearest cent as​ needed!

To find the future value of an annuity due of $900 semiannually for six years at 8% annual interest compounded semiannually, we need to use the annuity due formula:

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

where:
- $ P $ is the payment amount ($900)
- $ r $ is the interest rate per period (0.08 / 2 = 0.04)
- $ n $ is the number of periods (6 years $\times$ 2 = 12 periods)

From the table, we look for the value corresponding to 8% annual interest compounded semiannually (4% per period) for 12 periods. However, the table provided does not contain 8% (or 4% per period), so we will use the formula directly.

Plugging in the values:

\[ FV = 900 \times \left(\frac{(1 + 0.04)^{12} - 1}{0.04}\right) \times (1 + 0.04) \]

First, calculate $ (1 + 0.04)^{12} $:

\[ (1 + 0.04)^{12} = 1.601032 \]

Next, calculate the future value factor:

\[ \left(\frac{1.601032 - 1}{0.04}\right) = 15.0258 \]

Then multiply by $ (1 + 0.04) $:

\[ 15.0258 \times 1.04 = 15.6278 \]

Finally, multiply by the payment amount:

\[ FV = 900 \times 15.6278 = 14,065.02 \]

So the future value of the annuity is approximately $14,065.02.

The total investment over the 6 years is:

\[ \text{Total Investment} = 900 \times 12 = 10,800 \]

The earned interest is:

\[ \text{Earned Interest} = 14,065.02 - 10,800 = 3,265.02 \]

To summarize:
- The future value is approximately $14,065.02.
- The total investment was $10,800.00.
- The earned interest was approximately $3,265.02.

Would you like details or have any questions?

Here are 5 numbered questions you can ask:
1. How is the annuity due different from an ordinary annuity?
2. What is the formula for the future value of an ordinary annuity?
3. How do you calculate the interest rate per period for different compounding frequencies?
4. What are the steps to derive the future value formula for an annuity?
5. Can you explain the concept of present value in annuities?

**Tip:** Always check the compounding frequency to ensure you are using the correct interest rate per period when calculating annuities.

Learn how to calculate the future value of an annuity due with compound interest using detailed steps and formulas. This problem involves an annuity due of $900 semiannually for six years at 8% annual interest compounded semiannually, rounded to the nearest cent. Discover the total investment, earned interest, and key mathematical concepts. Suitable for college-level finance and mathematics students.

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