2026 RI Promo Q11

On 1 January 2025, Amy puts \(\$100\) into an ordinary account which pays no interest. In each calendar year, she deposits the same amount on the first day of every month. At the start of each new year, she increases her monthly deposit by \(\$30\) compared to the previous year. Thus she deposits \(\$100\) on the first day of every month in 2025, \(\$130\) on the first day of every month in 2026, and so on.

1. Find the year in which the total amount in the account will first exceed \(\$50\,000\). Hence find the month in which this happens.[6]

Amy’s mother, Beatrice, retired on 31 Dec 2024. On 1 Jan 2025, she puts \(\$500\,000\) into a savings account that pays a monthly interest of \(0.2\%\) credited on the last day of each month.

On the first day of each subsequent month, she withdraws \(\$x\) from the account for her daily expenses.

2. Show that the amount, in dollars, that Beatrice has in her account on the last day of the \(n\)th month (where January 2025 is the 1st month, February 2025 is the 2nd month, and so on) is
   \[1.002^n(500\,000-500x)+501x.\][3]
3. If \(x=4000\), in which year and month will Beatrice draw \(\$4000\) for the last time?[3]