a$)$ You deposit $$200$ monthly into an annuity with the goal of accumulating $$180,000$ after $30$ years. What annual rate, compounded monthly, is required to $?$accomplish this

b$)$ At retirement, a client has two payment options: a $20-$year annuity at $$50,000$ per year starting after $1$ year or a lump sum of $$500,000$ today. If the client's required rate of return on retirement fund investments is $6\%$ per year, which plan has the higher present $?$value and by how much