How much should be deposited into an account semiannually to have a balance of $9,500 in eight years if the rate is 1.25% compounded semiannually?

To determine how much should be deposited semiannually to achieve a balance of $9,500 in eight years, with a 1.25% interest rate compounded semiannually, we need to use the future value of an annuity formula.

### Future Value of an Annuity Formula:

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

Where:
- $FV$ is the future value (desired balance) = $9,500
- $P$ is the semiannual deposit amount (what we're solving for)
- $r$ is the semiannual interest rate
- $n$ is the total number of periods

### Step-by-Step Calculation:
1. **Convert the interest rate:**
   - Annual interest rate = 1.25%
   - Since interest is compounded semiannually, divide the rate by 2: 
     \[
     r = \frac{1.25\%}{2} = 0.00625
     \]

2. **Determine the number of periods:**
   - There are 2 periods per year, and the time frame is 8 years. So:
     \[
     n = 2 \times 8 = 16 \text{ periods}
     \]

3. **Plug the values into the formula and solve for $P$:**

\[
9500 = P \times \frac{(1 + 0.00625)^{16} - 1}{0.00625}
\]

Now, let's compute the value of $P$.The semiannual deposit needed to achieve a balance of $9,500 in eight years, with a 1.25% interest rate compounded semiannually, is approximately **$566.41**.

Would you like further details on any part of this calculation or process?

Here are 5 related questions to further your understanding:
1. How would the result change if the interest rate were higher?
2. What is the difference between simple and compound interest in this context?
3. How is the formula affected if the deposits were made quarterly instead of semiannually?
4. What happens if we extend the time period to 10 years instead of 8?
5. Can this formula be applied for different compounding periods?

**Tip:** When dealing with compound interest, always ensure that the interest rate and the number of compounding periods match the frequency of payments to get accurate results.

Discover how to calculate semiannual deposits required to achieve a balance of $9,500 in eight years with an interest rate of 1.25% compounded semiannually. This problem uses the future value of an annuity formula and covers compound interest concepts, suitable for students in Grades 10-12.

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