2022 HCI J1 BT Q11
On $31$st December $2021$, Mei Qian bought a house and took a housing loan of $\$500\text{}000$ from a bank that charges interest at $1.8\%$ per year, compounded on the outstanding loan amount on $1$st January of each year, starting with the year $2022$.
On $15$th January $2022$, Mei Qian made her first repayment of $\$1\text{}800$. On the fifteenth day of each subsequent month, she repaid the loan with a fixed monthly instalment of $\$1\text{}800$.
The outstanding loan amount at the end of $n$ years is denoted by $\${{L}_{n}}$ and ${{L}_{1}}$ is the outstanding loan on $31$st December $2022$.
(i)
Find ${{L}_{1}}$.
[2]
(ii)
Show that ${{L}_{n}}=A-B{{\left( 1.018 \right)}^{n}}$, where $A$ and $B$ are exact constants to be determined.
[3]
(iii)
Hence determine the least value of $n$ in which Mei Qian repays her loan.
[2]
(iv)
Find the date in which she made her last repayment and the amount she has to repay.
[3]
Also on $31$st December $2021$, Mei Qian signed up for a $35$-year investment scheme, GrowSmart, where the payout is made on the tenth day of each month. On $10$th January $2022$, the first payout is $\$200$. For each of the subsequent months, the payout will be $\$5$ more than the payout in the previous month. In this way, she would receive a payout of $\$205$ on $10$th February $2022$ and $\$210$ on $10$th March $2022$.
Mei Qian hoped that the monthly payouts from GrowSmart would one day be able to finance the monthly repayment for the housing loan completely.
(v)
Find the number of months required such that the monthly payout from GrowSmart could finance the monthly repayment for the housing loan completely.
[2]
(vi)
Find the total amount she received from the payouts over $35$ years.
[2]
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