Answer:
To find the monthly payment for a 3-year auto loan at 4% annual interest compounding monthly, we can use the formula for **monthly payment for a loan**:
```
M = P [ i(1 + i)^n ] / [ (1 + i)^n – 1]
```
where:
- M is the monthly payment
- P is the principal or amount borrowed
- i is the monthly interest rate
- n is the number of months
In this case, the principal or amount borrowed is $18,000 and the down payment is $2,000. Therefore, the principal amount that needs to be financed is $16,000.
The monthly interest rate can be calculated by dividing the annual interest rate by 12. In this case, the annual interest rate is 4%, so the monthly interest rate is 4% / 12 = 0.00333333.
The number of months can be calculated by multiplying the number of years by 12. In this case, the number of months is 3 years x 12 months/year = 36 months.
Now we can substitute these values into the formula:
```
M = $16,000 [ 0.00333333(1 + 0.00333333)^36 ] / [ (1 + 0.00333333)^36 – 1]
```
Simplifying this expression gives us:
```
M = $468.43
```
Therefore, Carmen's monthly payment will be **$468.43** .