# RE4 | google/gemini-3.6-flash | run 1

## 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 3 years, the final balance of the deposit will be **$1,157.63**. This calculation accounts for 5% interest compounded annually on both the original deposit and the accumulated interest.

## Compound Interest Calculation

The balance is calculated using the annual compound interest formula:

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

The variables for this calculation are:
- \( P = 1000 \) (Principal deposit)
- \( r = 0.05 \) (Annual interest rate)
- \( t = 3 \) (Time in years)
- \( A = \) Final balance

### Step-by-Step Work

1. Add 1 to the annual interest rate:
\( 1 + 0.05 = 1.05 \)

2. Raise the growth factor to the power of 3 years:
\( (1.05)^3 = 1.157625 \)

3. Multiply by the principal amount:
\( A = 1000 \times 1.157625 = 1157.625 \)

4. Round to the nearest cent:
\( A \approx 1157.63 \)

## Annual Growth Breakdown

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

Would you like to see how changing the compounding frequency (such as monthly or daily) would impact your total return?
