Suppose an individual makes an initial investment of $2400 in an account that earns 8%, compounded monthly, and makes additional contributions of $100 at the end of each month for a period of 12 years. After these 12 years, this individual wants to make withdrawals at the end of each month for the next 5 years (so that the account balance will be reduced to $0).

(a) How much is in the account after the last deposit is made?
(b) How much was deposited?
(c) What is the amount of each withdrawal?
(d) What is the total amount withdrawn?

To solve this problem, we'll break it down into steps:

Step 1: Calculate the future value of the initial investment plus the monthly contributions after 12 years.
Step 2: Calculate the monthly withdrawals needed to reduce the account balance to $0 over 5 years.
Step 3: Calculate the total amount withdrawn over the 5-year period.

Let's start with Step 1:

Step 1:
To calculate the future value (FV) of the investment after 12 years with monthly compounded interest, we can use the formula:

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

Where:
P = initial investment = $2400
r = annual interest rate = 8% = 0.08
n = number of compounding periods in a year = 12 (monthly)
t = number of years

Let's plug in the values and calculate the future value after 12 years:

FV = 2400(1 + 0.08/12)^(12*12)
FV = 2400(1 + 0.0066667)^(144)
FV ≈ 2400(1.006667)^144
FV ≈ 2400(1.979057)
FV ≈ $4750.58

So, after the last deposit is made, there will be approximately $4750.58 in the account.

Step 2:
To calculate the monthly withdrawals needed to reduce the account balance to $0 over 5 years, we need to consider the future value we calculated in Step 1. We'll use the future value as the starting balance (B), and use the formula:

B = P((1 + r)^t - 1)/r

Where:
P = monthly withdrawal amount (to be determined)
r = monthly interest rate = 8%/12 = 0.0066667
t = number of years = 5

We want the future value to be reduced to $0, so:

0 = P((1 + 0.0066667)^5 - 1)/0.0066667

We need to solve for P. Let's rearrange the equation and solve:

P((1 + 0.0066667)^5 - 1) = 0.0066667 * 0
P((1.006667)^5 - 1) ≈ 0
P ≈ 0/(1.006667^5 - 1)
P ≈ 0/0.033786

Since the denominator is very close to zero, we can assume the monthly withdrawal amount (P) approaches infinity. Therefore, it is not possible to withdraw an equal amount each month and reduce the account balance to $0 over 5 years.

Step 3:
Since it's not possible to withdraw equal amounts each month over 5 years to reduce the account balance to $0, we cannot determine the exact amount of each withdrawal (part c).

Step 4:
Since we couldn't calculate the exact amount for each withdrawal, we can't determine the total amount withdrawn (part d) either.

In summary:
a) The account balance after the last deposit is made is approximately $4750.58.
b) The total amount deposited is $2400 (initial investment) + $100 x 12 years x 12 months = $2400 + $14,400 = $16,800.
c) Not determinable.
d) Not determinable.

To answer these questions, we need to use the formula for compound interest:

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

Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times that interest is compounded per year
t = number of years the money is invested or borrowed for

Now let's apply this formula to solve each question:

(a) How much is in the account after the last deposit is made?

To find the account balance after the last deposit, we need to calculate the future value of the investment after the 12 years of monthly deposits. Then we can subtract the $100 contribution made in the 13th year.

Using the formula, we have:
P = $2400
r = 8% = 0.08
n = 12 (compounded monthly)
t = 12 years

A = 2400(1 + 0.08/12)^(12*12)
A ≈ $6488.35

Therefore, the account balance after the last deposit is made will be approximately $6488.35.

(b) How much was deposited?

To find the total amount deposited, we need to calculate the sum of all the monthly contributions made over 12 years.

The monthly contribution is $100, and it is made for 12 years (12 * 12 months).

Total contribution = $100 * 12 * 12
Total contribution = $14400

Therefore, the total amount deposited is $14400.

(c) What is the amount of each withdrawal?

To find the amount of each withdrawal, we need to divide the remaining account balance after the last deposit by the total number of withdrawals to be made (5 years * 12 months).

Remaining account balance = $6488.35
Total number of withdrawals = 5 years * 12 months = 60 months

Amount of each withdrawal = Remaining account balance / Total number of withdrawals
Amount of each withdrawal = $6488.35 / 60
Amount of each withdrawal ≈ $108.14

Therefore, the amount of each monthly withdrawal will be approximately $108.14.

(d) What is the total amount withdrawn?

To find the total amount withdrawn over the 5 years, we need to multiply the amount of each monthly withdrawal by the total number of withdrawals (5 years * 12 months).

Total amount withdrawn = Amount of each withdrawal * Total number of withdrawals
Total amount withdrawn = $108.14 * 60
Total amount withdrawn ≈ $6488.40

Therefore, the total amount withdrawn over the 5 years will be approximately $6488.40.