A government program that currently costs taxpayers $$4\mathrm{.}5$ billion per year is cut back by $20$ percent per year. $($a$)$ Write an expression for the amount budgeted for this program after years. $($b$)$ Compute the budgets for the first $4$ years. $($c$)$ Determine the convergence or divergence of the sequence of reduced budgets. If the sequence converges, find its limit