# RE4 | google/gemini-3.6-flash | run 2

## Prompt

You deposit $1,000 at 5% interest compounded annually. What is the balance after 3 years? Round to the nearest cent and show your work.

## Response

After depositing $1,000 at an annual interest rate of 5% compounded annually for 3 years, the final account balance will be $1,157.63. This calculation reflects earning interest on both the original principal and the interest accumulated over previous years.

## Compound Interest Calculation

To calculate the future value of a compound interest investment, use the standard compound interest formula:

\[ A = P(1 + r)^t \]

The variables are defined as follows:
- \( P \): Principal amount ($1,000)
- \( r \): Annual interest rate as a decimal (0.05)
- \( t \): Time in years (3)
- \( A \): Total accumulated amount

**Step 1: Substitute the values into the formula**
\[ A = 1000(1 + 0.05)^3 \]

**Step 2: Simplify the base**
\[ A = 1000(1.05)^3 \]

**Step 3: Evaluate the exponent**
\[ 1.05^3 = 1.157625 \]

**Step 4: Multiply by the principal**
\[ A = 1000 \times 1.157625 = 1157.625 \]

**Step 5: Round to the nearest cent**
Rounding 1157.625 to two decimal places yields **$1,157.63**.

## Yearly Balance Breakdown

| Year | Starting Balance | Interest Earned (5%) | Ending Balance |
| :--- | :--- | :--- | :--- |
| Year 1 | $1,000.00 | $50.00 | $1,050.00 |
| Year 2 | $1,050.00 | $52.50 | $1,102.50 |
| Year 3 | $1,102.50 | $55.13 | $1,157.63 |

Would you like to calculate the balance with a different compounding frequency, such as monthly or daily?
