2020 EJC P2 Q5

On 1 Jan 2019, Emma takes up a study loan of \({\$30000}\) from a bank that offers two repayment plans. Plan A charges a monthly instalment of \({\$600}\) at the beginning of every month and increases by \({\$50}\) for every subsequent month. There is no interest charged; instead, the bank charges a one-time administrative fee of \({\$2500}\) which is to be added to the study loan.

1. 1. Show that, after the \({{n}^{\mathrm{th}}}\) payment, she would have paid a total amount of \(\$\left(25{{n}^{2}}+575n\right)\).[2]
   2. How many payments will it take for Emma to fully repay her loan?[2]

Plan B charges a fixed monthly instalment of \(\$\,x\) which is to be paid at the beginning of each month, starting from 1 Jan 2019. There is no interest charged for the first month only. Thereafter, interest is charged at the end of each month on the outstanding amount at \(1\%\) per month, (i.e. interest will be first charged on 28 February 2019.)

2. 1. Show that the amount she owes after the fourth payment is \({{\left( 1.01 \right)}^{2}}\left( 30000 \right)-{{\left( 1.01 \right)}^{2}}\left( 2x \right)-1.01x-x\).[2]
   2. Use the formula for the sum of a geometric progression to find an expression, in terms of \(n\) and \(x\), the amount she owes after the \({{n}^{\mathrm{th}}}\) payment, for \(n\ge 2\).[2]
   3. If Emma intends to repay the loan fully after \(32\) payments, find the least value of \(x\).[2]
3. Using the value of \(x\) in part (ii)(c), determine which plan will be cheaper for Emma.[1]