N2008 P1 Q10 [Modified]

Joe puts \(\$10\) on \(1\) January \(2025\) into a bank account which pays compound interest at a rate of \({2\%}\) per month on the last day of each month. He puts a further \(\$10\) into the account on the first day of each subsequent month.

1. How much compound interest has his original \(\$10\) earned at the end of \({2}\) years?
2. Content for Let \({a_n}\) denote the total amount in the account at the end of \({n}\) months. Find the value of \({a_1}\) and the recurrence relation in the form \({a_{n+1} = f(a_n)}\).
3. With the aid of a graphing calculator, determine
   1. the total amount in the account at the end of \({2}\) years.
   2. the number of complete months for the total in the account to first exceed \(\$2000\).