You wish to purchased a house for $200,000 in 12 years. You can invest your money at 5%/a compounded semiannually. How much do you need to invest every 6 months, starting in 6 months, so that you will have $200,000 at the time of your last deposit? (Hint: You need to find the value for "a". )

To solve for the amount $a$ you need to invest every 6 months, we will use the future value of an ordinary annuity formula, which is:

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

where:
- $FV$ is the future value, which is $200,000
- $r$ is the interest rate per period
- $n$ is the number of periods
- $a$ is the annuity payment we need to find

Given:
- The future value $FV = 200,000$
- The annual interest rate is 5%, so the semiannual interest rate $r = \frac{5}{2 \times 100} = \frac{0.05}{2} = 0.025$
- The total number of periods $n = 12 \times 2 = 24$ (since payments are made semiannually for 12 years)

Plug these values into the formula and solve for $a$:

\[ 200,000 = a \times \frac{(1 + 0.025)^{24} - 1}{0.025} \]

First, calculate $(1 + 0.025)^{24}$:

\[ (1.025)^{24} \approx 1.6406 \]

Now, calculate the fraction:

\[ \frac{1.6406 - 1}{0.025} = \frac{0.6406}{0.025} = 25.624 \]

So, the equation becomes:

\[ 200,000 = a \times 25.624 \]

Solve for $a$:

\[ a = \frac{200,000}{25.624} \approx 7805.47 \]

Therefore, you need to invest approximately $7805.47 every 6 months to accumulate $200,000 in 12 years.

Would you like further details or have any questions?

Here are some related questions you might want to ask:
1. How do changes in the interest rate affect the amount you need to invest?
2. What is the difference between simple interest and compound interest?
3. How can you adjust the investment plan if you can make monthly deposits instead?
4. Can you explain the steps to derive the annuity formula?
5. What other factors should be considered when planning for a large future expense?

**Tip:** When dealing with compound interest, always ensure the interest rate and the number of periods are consistent with the compounding frequency.

Learn how to calculate the semiannual investment required to reach a future value of $200,000 using compound interest at 5% annually. This problem involves applying the future value of an ordinary annuity formula, considering semiannual compounding.

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