Question $6$: $20$ Marks

Consider a cheque account. Initially, the amount of R$10000$ is deposited on the account. At the end of each month, interest is added at the rate of $2\%$ per month to the amount of money that was on the account during the month. $A_n$ additional deposit of $N$ is added to the account at the end of each month, and the fixed amount R$500$ is withdrawn from the account. Let $A_n$ denote the amount of money in the account at the end of month $n,$ with $A_0$ denoting the original amount.

$(6\mathrm{.}1)$ Write down the difference equation for $A_n($that is$,$ an expression for $A_{n+1}$ in terms of $A_n).$ Write also down the initial value, $A_0.$

$(6\mathrm{.}2)$ How should the value $N$ be chosen such that there will always be money on the account? Justify your answer!