Investment and withdrawal model

On the day of her birth, \(1^{\mathrm{st}}\) January \(1998\), Mary's grandparents invested \(\$x\) in a savings account. They continued to deposit \(\$x\) on the first day of each month thereafter.

The account paid a fixed rate of \(0.4\)% interest per month. The interest was calculated on the last day of each month and added to the account.

Let \(\$A_n\) be the amount in Mary's account on the last day of the \(n^{\mathrm{th}}\) month, immediately after the interest has been added.

1. Find an expression for \(A_1\) and show that \(A_2 = 1.004^2 x + 1.004 x\).
2. 1. Write down a similar expression for \(A_3\) and \(A_4\).
   2. Hence show that the amount in Mary's account the day before she turned \(10\) years old is given by \(25(1.004^{120} - 1)x\).
3. Write down an expression for \(A_n\) in terms of \(x\) on the day before Mary turned \(18\) years old showing clearly the value of \(n\).
4. Mary's grandparents wished for the amount in her account to be at least \(\$20\,000\) the day before she was \(18\). Determine the minimum value of the monthly deposit \(\$x\) required to achieve this. Give your answer correct to the nearest dollar.
5. As soon as Mary was \(18\), she decided to invest \(\$15\,000\) of this money in account of the same type earning \(0.4\)% interest per month. She withdraws \(\$1000\) every year on her birthday to buy herself a present. Determine how long it will take until there is no money in the account.