The Perpetual Life Insurance Co$.$ is trying to sell you an investment policy that will pay you and your heirs $$20,000$ per year forever. $($a$)$ If the discount rate on this investment is $4\mathrm{.}5$ percent, how much will you pay for the policy? $($b$)$ Suppose the Perpetual Life Insurance Co$.$ told you the policy costs $$500,000.$ At what discount rate would this be a fair deal?