Suppose that a small town has a pension fund that is expected to make annual payments totaling $$500,000$ to its local government retirees. The fund must be able to sustain such payments in perpetuity; that is$,$ forever. The easiest way for this to happen is if the town is able to set aside $$500,000$ every year to pay its retirements.

Another possibility to sustain the fund would be to plan for the future. The city could allocate and invest money for its pension fund from the time the first local government employee is hired until the time he or she retires $30$ years later. To simplify our calculations, assume that the city will set aside an amount d in year $1$ to pay for the first year of retirement for all local government retirees $30$ years later. The council believes it can earn $7\%$ annual interest on the account.

$4.$ Calculate the amount the city must reserve that first year. Does the $30-$year plan benefit the city? Explain.