N2023 P1 Q11

Wei is saving for a property that she intends to buy in the future. Wei needs to save a minimum of \(\$50 \,\, 000\). She saves regularly in an account which offers no interest. She makes an initial deposit on 31 January 2021 of \(\$a\). Each subsequent month, she deposits \(\$50\) more than she deposited in the previous month. Her final deposit is made on 31 December 2023.

1. Find, to the nearest cent, the smallest value of \(a\) so that she saves at least \(\$50\,\,000\).[2]

Wei has arranged to purchase a property on 1 January 2024 with the aid of a loan of \(\$400\,\,000\). The terms of the loan are that interest of \(0.1\%\) is added to the amount owing at the start of every month, with the first interest amount added on 1 January 2024. Wei makes a monthly repayment of \(\$x\) at the end of every month, with the first repayment on 31 January 2024.

2. Show that the amount, in dollars, that Wei owes at the end of \(n\) months is
   \(400\,\,000\times {{1.001}^{n}}-1000x\left( {{1.001}^{n}}-1 \right)\).[3]
3. 1. Suppose that the loan is repaid in \(360\) monthly repayments. Find the value of the monthly repayment to the nearest cent.[2]
   2. Use your answer to part (c)(i) to calculate the total interest paid on the loan in this case.[1]
4. 1. Now suppose Wei decides to repay the loan at the rate of \(\$1600\) per month for \(k\) months, plus a final monthly repayment of \(\$y\), where \(y<1600\). Find \(k\) and \(y\).[4]
   2. Hence find the total saving to Wei by making the repayments in part (d)(i) rather than the repayments in part (c)(i), giving your answer to \(4\) significant figures.[1]