At the beginnig of each of the years 2023 and 2024, Amy deposits $10000 into a bank. The interest is compounded annually by the bank. If she can receive $20800 at the end of the year 2024 and there is no more deposit afterward, find the amount she can receive at the end of the year 2026 correct to the nearest $100

Let's break down the problem to determine the amount Amy can receive at the end of the year 2026.

### Step 1: Determine the Interest Rate
Amy deposited $10,000 at the beginning of both 2023 and 2024, and by the end of 2024, she has $20,800. We need to find the annual interest rate $ r $.

Her total deposit by the end of 2024 is $20,000. Let the interest rate be $ r $.

For the amount at the end of 2024, we consider the interest compounded annually:

1. The amount from the deposit made at the beginning of 2023 would be compounded for 2 years:
   \[
   10,000 \times (1 + r)^2
   \]
2. The amount from the deposit made at the beginning of 2024 would be compounded for 1 year:
   \[
   10,000 \times (1 + r)
   \]

The total amount at the end of 2024 is:
\[
10,000 \times (1 + r)^2 + 10,000 \times (1 + r) = 20,800
\]

We can factor out $ 10,000 \times (1 + r) $ from both terms:
\[
10,000 \times (1 + r) \times \left[(1 + r) + 1\right] = 20,800
\]

Simplifying further:
\[
10,000 \times (1 + r) \times (2 + r) = 20,800
\]

Divide both sides by 10,000:
\[
(1 + r) \times (2 + r) = 2.08
\]

Expanding and rearranging the quadratic equation:
\[
r^2 + 3r - 0.08 = 0
\]

### Step 2: Solve for $ r $
We solve this quadratic equation using the quadratic formula:
\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where $ a = 1 $, $ b = 3 $, and $ c = -0.08 $.

\[
r = \frac{-3 \pm \sqrt{3^2 + 0.32}}{2}
\]
\[
r = \frac{-3 \pm \sqrt{9.32}}{2}
\]
\[
r \approx \frac{-3 \pm 3.05}{2}
\]

Taking the positive root:
\[
r \approx \frac{0.05}{2} = 0.025
\]

So the interest rate $ r $ is approximately 2.5%.

### Step 3: Calculate the Amount in 2026
Now, we calculate the amount Amy can receive at the end of 2026. At the end of 2024, she has $20,800, and this amount will compound for 2 more years at an interest rate of 2.5%.

\[
\text{Amount in 2026} = 20,800 \times (1.025)^2
\]
\[
\text{Amount in 2026} \approx 20,800 \times 1.050625
\]
\[
\text{Amount in 2026} \approx 21,851.00
\]

Rounded to the nearest $100, the amount Amy can receive at the end of 2026 is **$21,900**.

Would you like further details, or do you have any questions?

Here are 8 follow-up questions to explore:

1. What would happen to Amy's returns if the interest were compounded semi-annually instead of annually?
2. How would the final amount change if Amy made additional deposits in 2025?
3. How does compound interest compare to simple interest in terms of returns over multiple years?
4. What if the interest rate was variable instead of fixed at 2.5%?
5. Can you solve for the interest rate using a different method, such as trial and error?
6. What if the deposits were made at the end of each year instead of the beginning?
7. How does inflation affect the real value of Amy's returns?
8. What strategies could Amy use to maximize her returns over the long term?

**Tip:** When dealing with compound interest, always double-check the compounding period, as different periods can significantly affect the outcome.

Explore how to calculate compound interest with annual deposits using a detailed step-by-step solution. This problem involves determining future amounts based on annual deposits and compound interest rates. Suitable for advanced high school students and learners interested in financial mathematics.

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