2013 DHS P1 Q7

During a promotion, a bank offers an interest-free loan for the first three months. Thereafter, interest is charged at the end of each month on the outstanding amount at 1% per month. John borrows \(\$30000\) from the bank and pays back a fixed amount of \(\$x\) at the beginning of each month, starting from the first month.

1. Find in terms of \(x\), the amount he owes after the third payment.[2]
2. Show that the amount he owes after the fifth payment is
   \({{1.01}^{2}}\left( 30000 \right)-{{1.01}^{2}}\left( 3x \right)-1.01x-x.\)[2]
3. Find the amount he owes after the \({{n}^{th}}\) payment, simplifying your answer in terms of \(n\) and \(x\) using the sum of a geometric progression formula.[3]
4. If the loan is fully repaid after \(60\) payments, find the amount of each monthly payment.[2]