$5.$ When he turned $42,$ Jim started to contribute to his retirement account by making an annual deposit of $$2500,$ which is matched by his employer by $150\%$ in an ordinary annuity bearing $9(1)/(2)\%$ interest compounded semiannually.

a$.$ How much will he have available for him when he retires at $65?$

b$.$ If Jim wishes to have a total of $$600,000$ in his IRA when he retires, How much should his $($and his employer$$s$)$ annual contribution be$?$ Consider the same $($r$)$ and $($n$)$ of question# $5-$a$.$

c$.$ Consider question #$5-$a$,$ and suppose that Jim$$s contribution is $$2000$ but matched by his employer as double $($two for one$).$ Also suppose that Jim$$s goal is to accumulate half a million dollar in his IRA. At what age should his retirement be to achieve that goal, given that his account bears $10\%$ interest compounded semiannually?$6.$ For the following $6-$a$,6-$b$,$ and $6-$c$,$ recalculate what you did in $5-$a through $5$c assuming that the retirement contributions are deposited in an annuity due account.