An individual saves $2000 in a bank account at the beginning of each year for 8 years. No further
savings or withdrawals are made from the account. Determine the total amount saved if the annual
interest rate is 6% compounded semi-annually.

To determine the total amount saved in the account after 8 years with an annual interest rate of 6% compounded semi-annually, we can use the formula for the future value of an annuity:

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

where:
- $ P $ is the annual payment (in this case, $2000),
- $ r $ is the annual interest rate (6% or 0.06),
- $ n $ is the number of compounding periods per year (2 for semi-annual),
- $ t $ is the number of years (8 years).

### Step 1: Identify the values
- $ P = 2000 $
- $ r = 0.06 $
- $ n = 2 $
- $ t = 8 $

### Step 2: Calculate the future value (FV)
Now, substitute these values into the formula:

\[
FV = 2000 \times \left( \frac{(1 + 0.06/2)^{2 \times 8} - 1}{0.06/2} \right) \times \left(1 + \frac{0.06}{2} \right)
\]

Let's break this down further:

1. **Calculate $ \frac{r}{n} $:**
   \[
   \frac{0.06}{2} = 0.03
   \]
2. **Calculate $ (1 + 0.03)^{16} $ (since $ nt = 16 $):**
   \[
   (1.03)^{16}
   \]
3. **Subtract 1 and divide by 0.03:**
   \[
   \frac{(1.03)^{16} - 1}{0.03}
   \]
4. **Multiply the result by $ P $ (2000) and $ (1 + 0.03) $:**
   \[
   2000 \times \text{result} \times 1.03
   \]

I'll compute the exact value now.

The total amount saved in the bank account after 8 years, with a 6% interest rate compounded semi-annually, is approximately **$41,523.18**.

Would you like further details or have any questions?

Here are 5 related questions that might interest you:
1. How does changing the interest rate affect the total savings?
2. What if the compounding was done quarterly instead of semi-annually?
3. How would the savings amount differ if the individual saved $3000 each year instead?
4. What happens if the individual withdraws money after the 4th year?
5. How would the total savings differ if no interest was applied?

**Tip:** Understanding how different compounding intervals (e.g., quarterly vs. semi-annually) impact the final amount is crucial when planning long-term savings.

An individual saves $2000 in a bank account at the beginning of each year for 8 years. No further savings or withdrawals are made from the account. Determine the total amount saved if the annual interest rate is 6% compounded semi-annually.

Learn how to calculate the total savings when $2000 is deposited annually for 8 years with a 6% interest rate compounded semi-annually. This problem explores key financial concepts such as compound interest and annuities, making it ideal for students in Grades 10-12.

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