To solve this problem, we need to use the formula for compound interest. The formula for compound interest is:

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

Where:
- $ A $ is the amount of money accumulated after n years, including interest.
- $ P $ is the principal amount (the initial amount of money).
- $ r $ is the annual interest rate (decimal).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the time the money is invested for in years.

Given:
- $ P = 3000 $ dollars
- $ r = 0.02 $ (2% annual interest rate)
- $ n = 12 $ (compounded monthly)
- $ t = 5 $ years

First, we need to convert the annual interest rate to a monthly interest rate by dividing by 12:

$$ \frac{r}{n} = \frac{0.02}{12} \approx 0.0016667 $$

Next, we substitute the values into the compound interest formula:

$$ A = 3000 \left(1 + 0.0016667\right)^{12 \times 5} $$

Calculate the exponent:

$$ 12 \times 5 = 60 $$

Now, calculate the base:

$$ 1 + 0.0016667 = 1.0016667 $$

Raise the base to the power of 60:

$$ 1.0016667^{60} \approx 1.104713 $$

Finally, multiply by the principal amount:

$$ A = 3000 \times 1.104713 \approx 3314.14 $$

So, the final value after 5 years, with interest compounded monthly, is approximately $3314.14.

Question

To solve this problem, we need to use the formula for compound interest. The formula for compound interest is:

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

Where:
- $ A $ is the amount of money accumulated after n years, including interest.
- $ P $ is the principal amount (the initial amount of money).
- $ r $ is the annual interest rate (decimal).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the time the money is invested for in years.

Given:
- $ P = 3000 $ dollars
- $ r = 0.02 $ (2% annual interest rate)
- $ n = 12 $ (compounded monthly)
- $ t = 5 $ years

First, we need to convert the annual interest rate to a monthly interest rate by dividing by 12:

$$ \frac{r}{n} = \frac{0.02}{12} \approx 0.0016667 $$

Next, we substitute the values into the compound interest formula:

$$ A = 3000 \left(1 + 0.0016667\right)^{12 \times 5} $$

Calculate the exponent:

$$ 12 \times 5 = 60 $$

Now, calculate the base:

$$ 1 + 0.0016667 = 1.0016667 $$

Raise the base to the power of 60:

$$ 1.0016667^{60} \approx 1.104713 $$

Finally, multiply by the principal amount:

$$ A = 3000 \times 1.104713 \approx 3314.14 $$

So, the final value after 5 years, with interest compounded monthly, is approximately $3314.14.

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