This problem has to do with exponential models. The question says, you deposit $1600 in a bank account. Find the balance after 3 years for each of the following situations. The first one says:

1. The account pays 2.5% annual interest compounded monthly.
2. The account pays 1.75% annual interest compounded yearly.
3. The account pays 4% annual interest compounded yearly.
For the first one, I think that I'm supposed to solve the equation this way: $1600 (1+.025/12 which is 1.0028)^36, since it's three years. I tried doing this same formula for the other two, except changing monthly to quarterly and yearly, but it doesn't work out. The other two are smaller numbers than the first, which doesn't make sense because it should increase. Can someone please help me?

Your first number should have been

1600*(1.00208333)^36 = $1724.48
You left out a 0 after the 2 in the 1.0020833, and you should not have rounded off to five significant figures. You need a total of at least six significant figures to keep track of the principal accurately.

Your third number should be
1600(1.04)^3 = $1799.78, which is not less than the first.

The second case of course returns less because of the very low 1.75% interest rate.

Thank you.

To find the balance after 3 years for each situation, let's break it down step by step:

1. The account pays 2.5% annual interest compounded monthly.
In this case, you are correct. The formula to calculate the balance after 3 years with monthly compounding is:
Balance = $1600 * (1 + 0.025/12)^(12 * 3)
Simplifying the equation, we have:
Balance = $1600 * (1.0020833)^36
Calculating this, you should get a balance of approximately $1708.11.

2. The account pays 1.75% annual interest compounded yearly.
To calculate the balance in this case, use the formula:
Balance = $1600 * (1 + 0.0175)^3
Calculating this, you should get a balance of approximately $1702.13.

3. The account pays 4% annual interest compounded yearly.
Similarly, use the formula:
Balance = $1600 * (1 + 0.04)^3
Calculating this, you should get a balance of approximately $1792.32.

It's important to note that the interest rate and compounding period affect the final balance. In this case, situation 1 pays the highest interest rate with monthly compounding, which is why the balance is highest. The other situations have lower interest rates or less frequent compounding, resulting in lower balances.

To calculate the balance after a certain period of time using exponential models, you need to use the formula:

A = P(1 + r/n)^(nt)

Where:
A = the final amount (balance) in the account
P = the principal amount (initial deposit)
r = annual interest rate (in decimal form)
n = the number of compounding periods per year
t = the number of years

Now let's calculate the balance for each of the given situations:

Situation 1: Account pays 2.5% annual interest compounded monthly
Using the formula:
A = $1600(1 + 0.025/12)^(12*3)
Simplifying the calculation:
A = $1600(1.0020833333)^36
A ≈ $1731.16

Situation 2: Account pays 1.75% annual interest compounded yearly
Using the formula:
A = $1600(1 + 0.0175/1)^(1*3)
Simplifying the calculation:
A = $1600(1.0175)^3
A ≈ $1639.26

Situation 3: Account pays 4% annual interest compounded yearly
Using the formula:
A = $1600(1 + 0.04/1)^(1*3)
Simplifying the calculation:
A = $1600(1.04)^3
A ≈ $1731.84

Now, let's compare the balances for each situation:

Situation 1: $1731.16
Situation 2: $1639.26
Situation 3: $1731.84

As you can see, the balance for situation 1 is higher than the others due to the additional compounding that occurs monthly. Additionally, the balance for situation 3 is slightly higher than situation 1 because of the higher interest rate (4% compared to 2.5%).

It's essential to use the correct formula and ensure you input the values accurately to get the correct results.