Application of GP to financial problems

Phil takes out a bank loan of \(\$150\,000\) to buy a house, at an annual rate of \(3.5\%\).

The interest is calculated at the end of each year and added to the amount outstanding.

1. Find the amount Phil would owe the bank after \(20\) years. Give your answer to the nearest dollar.[3]

To pay off the loan, Phil makes an annual deposit of \$ \(P\) at the end of every year in a savings account, paying an annual interest rate of \(2\)% . He makes his first deposit at the end of the first year after taking the loan.

2. Show that the total value of Phil's savings after \(20\) years is \[\frac{(1.02^{20} - 1)P}{1.02 - 1}\][4]
3. Given that Phil's aim is to own the house after \(20\) years, find the value of \(P\) to the nearest dollar.[2]

4. David visits a different bank and makes a single deposit \$ \(Q\), the annual interest rate being \(2.8\)% .
   1. David wishes to withdraw \$ \(5000\) at the end of each year for a period of \(n\) years. Show that an expression for the minimum value of \(Q\) is \[\frac{5000}{1.028} + \frac{5000}{1.028^2} + \cdots + \frac{5000}{1.028^n}\]
   2. Hence, or otherwise, find the minimum value of \(Q\) that would permit David to withdraw annual amounts of \$ \(5000\) indefinitely. Give your answer to the nearest dollar.