2014 IJC Promo Q10

In Gringotts Bank, interest is added to an account at the end of each year at a fixed rate of \(2\)% of the amount in the account at the beginning of that year.

1. On 1 January 2014, Mr Weasley opened a savings account in the bank. He decided to deposit \(\$R\) into his account and he would deposit the same amount on the first day of each subsequent year. In order to support his family financially, he needs to withdraw the interest as soon as it has been added. Find the total amount of interest Mr Weasley would receive on 31 December 2040.[4]
2. On the same day, Mr Potter also opened a savings account and deposited \(\$H\) into it. However for his case, he has decided that he will not withdraw any money out of his account and that he will deposit \(\$1000\) on the first day of each subsequent year. Let \({{S}_{n}}\) denote the total amount in his account at the end of \(n\) years.
   1. Write down the expression for S1 in terms of H and show that \({{S}_{n}}={{\left( 1.02 \right)}^{n}}H+51000\left( {{1.02}^{n-1}}-1 \right)\) .[5]
   2. Find the value of \(n\) for which the amount in his account exceeds \(\$25000\) for the first time, given that he deposited \(\$2000\) on 1 January 2014.[3]