Question $4$: $20$ Marks

Your grandparents have an annuity. The value of the annuity increases each month by an automatic deposit of $4\%$ interest on the previous month's balance. Your grandparents withdraw R$8000$ each month for living expenses. Currently, they have R$150000$ in the annuity. Let $A_n$ represent the funds in the annuity at the end of the month $n,$ with $A_0$ being the initial principal amount. Discuss the outcome of an annuity based on a comparison between the equilibrium value and the initial principal amount. Also, use a suitable model to compute when will the annuity be depleted.