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Melinda visits a different bank and makes a single deposit of $Q, the annual interest rate being 3.5%. Melinda wishes to withdraw $8000 at the end of each year for a period of n years. Show that an expression for the minimum value of Q is 8000/1.035+8000/1.035²+...+8000/(1.035)^(n)​

User Sonorita
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Answer:

To find the expression for the minimum value of Q, we need to consider the present value of the withdrawals that Melinda wishes to make over a period of n years.

The formula to calculate the present value of a series of future cash flows is:

Present Value (PV) = CF1 / (1 + r)^1 + CF2 / (1 + r)^2 + ... + CFn / (1 + r)^n

Where:

PV = Present Value

CF1, CF2, ..., CFn = Cash flows in each period (withdrawals)

r = Interest rate per period (annual interest rate / number of periods per year)

In this case, Melinda wants to withdraw $8000 at the end of each year for n years, and the annual interest rate is 3.5%.

So, we can rewrite the formula for Melinda's case as:

PV = 8000 / (1 + 0.035)^1 + 8000 / (1 + 0.035)^2 + ... + 8000 / (1 + 0.035)^n

Now, let's simplify this expression further:

PV = 8000 / 1.035^1 + 8000 / 1.035^2 + ... + 8000 / 1.035^n

Now, observe that the denominators follow a geometric sequence with a common ratio of 1/1.035:

1, 1/1.035, (1/1.035)^2, ..., (1/1.035)^(n-1), (1/1.035)^n

Using the formula for the sum of a geometric sequence, we can rewrite the expression as:

PV = 8000 * [(1 - (1/1.035)^n) / (1 - 1/1.035)]

Now, let's simplify the denominator:

1 - 1/1.035 = (1.035 - 1) / 1.035 = 0.035 / 1.035 = 0.035 / 0.035 = 1

So, the expression becomes:

PV = 8000 * [(1 - (1/1.035)^n) / 1]

PV = 8000 * (1 - (1/1.035)^n)

Therefore, the expression for the minimum value of Q is 8000 * (1 - (1/1.035)^n), which represents the present value of the withdrawals Melinda wishes to make over n years with an annual interest rate of 3.5%.

User MoLow
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