<p>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.</p>
<p>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.</p>
<p>Let&nbsp;$\$A_n$ be the amount in Mary's account on the last day of the $n^{\mathrm{th}}$ month, immediately after the interest has been added.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find an expression for&nbsp;$A_1$ and show that&nbsp;$A_2 = 1.004^2 x + 1.004 x$.</span></div></li><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down a similar expression for&nbsp;$A_3$ and $A_4$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence show that the amount in Mary's account the day before she turned $10$ years old is given by $25(1.004^{120} - 1)x$.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Write down an expression for&nbsp;$A_n$ in terms of $x$ on the day before Mary turned $18$ years old showing clearly the value of $n$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">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.</span></div></li><li><div class="list-item-content"><span class="list-item-text">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.</span></div></li></ol>

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.

Find an expression for $A_1$ and show that $A_2 = 1.004^2 x + 1.004 x$.

Write down a similar expression for $A_3$ and $A_4$.

Hence show that the amount in Mary's account the day before she turned $10$ years old is given by $25(1.004^{120} - 1)x$.

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 $n$.

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.

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.

Investment and withdrawal model

Investment and withdrawal model