How much should Michelle have in a savings account that is earning 3.50% compounded quarterly, if he plans to withdraw $1,500 from this account at the end of every quarter for 5 years? Round to the nearest cent

To determine how much Michelle should have in a savings account that earns 3.50% interest compounded quarterly, and from which she plans to withdraw $1,500 at the end of every quarter for 5 years, we need to use the formula for the present value of an annuity. The formula is:

$PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r$

Where:

• $PV$ is the present value (the amount Michelle needs to have in the account initially).
• $PMT$ is the payment amount per period ($1,500).
• $r$ is the interest rate per period.
• $n$ is the total number of periods.

First, we need to determine the values for $r$ and $n$:

1. The annual interest rate is 3.50%, so the quarterly interest rate $r$ is: $r = \frac{3.50\%}{4} = \frac{0.035}{4} = 0.00875$

2. The total number of periods $n$ is: $n = 5 \text{ years} \times 4 \text{ quarters/year} = 20 \text{ quarters}$

Now, we can plug these values into the formula:

$PV = 1500 \times \left(1 - (1 + 0.00875)^{-20}\right) / 0.00875$

First, calculate $(1 + 0.00875)^{-20}$:

$(1 + 0.00875)^{-20} \approx 0.8356$

Next, calculate $1 - 0.8356$:

$1 - 0.8356 = 0.1644$

Now, divide by the quarterly interest rate $r$:

$\frac{0.1644}{0.00875} \approx 18.7943$

Finally, multiply by the payment amount $PMT$:

$PV = 1500 \times 18.7943 \approx 28191.45$

Therefore, Michelle should have approximately $28,191.45 in the savings account.