$4.$ Patrick Mahomes is interested in buying a condo and has saved $$100,000$ for the down payment. His plans call for making additional monthly deposits into an investment account over the next $36$ months. Patrick wants to make the purchase $48$ months from today and wants to have $$200,000$ saved up for the down payment.

a$.$ What is the amount of each of the additional payments he must make for his plan to work out? Assume Patrick can earn $5\%$ annual return in his investment account.

b$.$ If Patrick takes out a $30-$year, $$800,000$ mortgage at $8\%,$ what will his mortgage payments be$?$

c$.$ Prepare an amortization schedule to prove that your answer is right, and his mortgage will be paid off in $30$ years.

d$.$ If Patrick takes out a $20-$year, $$800,000$ mortgage at $6\%,$ what will his mortgage payments be$?$

e$.$ How much would Patrick save if he took out a $20-$year mortgage versus a $30-$year mortgage?