2024 ASRJC Promo Q12

At a particular bank, a savings account pays an interest of \(0.9\%\) per month on the last day of each month. On 1 January 2024 Mr Monie put an initial deposit of \(\$10\,000\) into the savings account and continues to make a deposit of \(\$810\) at the start of each subsequent month.

1. Find the total amount in Mr Monie’s savings account at the end of the \(2\) months.[1]
2. Show that the total amount in the savings account at the end of the \(n\)th month is given by \(\$100\,900{{\left(1.009\right)}^{n-1}}-\$90810\).[3]
3. On what date did the value of Mr Monie’s account first exceed \(\$30\,000\)?[4]

Mr Pey has \(\$30\,000\) and is considering whether to keep this amount (with no further deposit) in the savings account to earn interest or invest this amount in another financial product. This financial product gives a disbursement of \(\$100\) at the end of the first month. At the end of every subsequent month, the financial product gives \(\$10\) more than the previous month. For example, at the end of first month, Mr Pey will have \(\$30\,100\) and at the end of two months, he will have \(\$30\,210\), and so on.

4. Find, in terms of \(N\), the total amount Mr Pey will have at the end of \(N\) months if he invests in the financial product.[2]
5. Given that \( 0 < N < 400\), find the range of values of \(N\) such that investing in the financial product is more profitable than keeping the money in the savings account.[2]