$($Growing Series Problem$)($a$)$ What amount must be deposited today into an account offering a stated annual rate of $10\mathrm{.}2\%$ compounded semiannually, if$,$ beginning twelve years from today, one wishes to make the following series of seven growing annual withdrawals? End$-$of$-$Year Withdrawal $12$ $$1,000(1\mathrm{.}054601)413$ $$1,000(1\mathrm{.}054601)514$ $$1,000(1\mathrm{.}054601)615$ $$1,000(1\mathrm{.}054601)716$ $$1,000(1\mathrm{.}054601)817$ $$1,000(1\mathrm{.}054601)918$ $$1,000(1\mathrm{.}054601)10($b$)$ What would be the deposit amount if the series continued for $50$ years to the end of year $61($last withdrawal $=$ $$1,000(1\mathrm{.}04)53)?($c$)$What would be the deposit amount if the series continues forever?