A business owner opens a new investment account with a $25,000 deposit:
 
The investment account earns 3.5% interest, compounded annually.
The business owner does not plan to make any deposits or withdrawals.
How much interest will the investment account earn at the end of the first and second years?

To calculate the interest earned on the investment account at the end of the first and second years, we use the **compound interest formula**:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ is the amount of money in the account after interest.
- $ P $ is the principal (initial deposit), which is $25,000.
- $ r $ is the annual interest rate (3.5% or 0.035).
- $ n $ is the number of times the interest is compounded per year (since it’s compounded annually, $ n = 1 $).
- $ t $ is the time in years.

Now, let’s calculate the amount in the account at the end of the first and second years, and then find the interest earned by subtracting the initial deposit from the final amounts.

### **Interest after 1 year:**

\[
A_1 = 25000 \left(1 + \frac{0.035}{1}\right)^{1 \times 1}
\]

\[
A_1 = 25000 \times (1.035)
\]

\[
A_1 = 25,875
\]

The amount after 1 year is **$25,875**. The interest earned is:

\[
\text{Interest}_1 = A_1 - P = 25,875 - 25,000 = 875
\]

So, the interest earned after 1 year is **$875**.

### **Interest after 2 years:**

\[
A_2 = 25000 \left(1 + \frac{0.035}{1}\right)^{1 \times 2}
\]

\[
A_2 = 25000 \times (1.035)^2
\]

\[
A_2 = 25000 \times 1.071225
\]

\[
A_2 = 26,780.625
\]

The amount after 2 years is **$26,780.63** (rounded to 2 decimal places). The interest earned is:

\[
\text{Interest}_2 = A_2 - P = 26,780.63 - 25,000 = 1,780.63
\]

So, the total interest earned after 2 years is **$1,780.63**.

### Summary:
- **Interest after 1 year**: $875
- **Interest after 2 years**: $1,780.63

Would you like more details on any step or additional clarifications?

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Here are 5 related questions to extend this concept:
1. How would the interest change if it were compounded monthly instead of annually?
2. What if the interest rate was 4% instead of 3.5%?
3. How much total interest would be earned after 5 years?
4. What would happen if the owner made an additional deposit after the first year?
5. How would the calculation change if the interest was simple instead of compound?

**Tip**: Compound interest grows faster over time because you earn interest on both the initial deposit and the accumulated interest.

A business owner opens a new investment account with a $25,000 deposit: The investment account earns 3.5% interest, compounded annually. The business owner does not plan to make any deposits or withdrawals. How much interest will the investment account earn at the end of the first and second years?

Learn how to calculate compound interest on a $25,000 investment at an annual interest rate of 3.5%. This problem demonstrates the compound interest formula, showing how much interest is earned after 1 and 2 years. Suitable for high school students studying exponential growth and finance.

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