2025 DHS Promo Q12

Desmond is saving to buy a car, and he needs to save a minimum of $40 000. Desmond saves regularly in an account which offers no interest. He made an initial deposit on 31 January 2022 of \($m\). In each subsequent month, he deposits $30 more than he deposited in the previous month. His final deposit will be made on 31 December 2025.

1. Find, to the nearest dollar, the smallest value of \(m\) so that he saves at least $40 000.[2]

Desmond has arranged to purchase the car on 1 January 2026 with the aid of a loan of $160 000. The terms of the loan are that interest of \(0.2\%\) is added to the amount owing at the start of every month, with the first interest amount added on 1 January 2026. He will make a monthly repayment of \($x\) at the end of every month, with the first repayment on 31 January 2026.

2. Show that the amount, in dollars, that Desmond owes at the end of \(n\) months is[3]
   \((1.002)^n(160\ 000-500x)+500x.\)
3. 1. Suppose that the loan is to be repaid in 60 monthly repayments. Find the value of the monthly repayment to the nearest dollar.[2]
   2. Use your answer in part (c)(i), calculate the total interest paid on the loan in this case.[1]
4. Desmond decides to repay the loan at the rate of $2000 per month for \(k\) months and a final monthly repayment of \($y\), where \(y<2000\). Find the values of \(k\) and \(y\).[4]