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 $0.5\%$.
(i)
Show that the outstanding amount at the beginning of ${{n}^{\text{th}}}$ 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.
(ii)
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.
(iii)
Show that the total amount that Mrs A saves by the beginning of ${{n}^{\text{th}}}$ month is $975n+25{{n}^{2}}$.
[2]
(iv)
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]
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