2022 NYJC Promo Q10

On 1 January 2021, Mr A takes a housing loan of \(\$300\,000\) from a bank.

On the same day, Mr A repays the bank \(\$x\), and on the first day of each subsequent month, he repays another \(\$x\). The interest rate is \(0.5\%\) per month, so that on the last day of each month, the outstanding loan amount will be increased by \(0.5\%\).

1. Show that the outstanding amount at the beginning of \({{n}^{\mathrm{th}}}\) month, after Mr A has repaid the bank, is \({{1.005}^{n-1}}\left( 300000 \right)-200x\left( {{1.005}^{n}}-1 \right)\).)[3]

Mr A repays the bank \(\$2\,500\) on the first day of each month.

2. On what date will Mr A fully pay off the housing loan?[3]

Mrs A starts saving \(\$1000\) on 1 January 2021. On the first day of each subsequent month, she is going to save \(\$50\) more than the previous month. For example, she will save \(\$1050\) on 1 February 2021 and \(\$1100\) on 1 March 2021.

3. Show that the total amount that Mrs A saves by the beginning of \({{n}^{\mathrm{th}}}\) month is \(975n+25{{n}^{2}}\).
4. To help Mr A with the housing loan, Mrs A decides to use her savings to make a one-time payment to fully repay the outstanding loan. Mrs A will make the one-time repayment once her savings is more than the amount that Mr A owes the bank.
   Determine the total amount of savings that Mrs A will use in the one-time repayment, giving your answer to the nearest cent.[4]