$1)$ Cameron purchased an annuity that had an interest rate of $2\mathrm{.}75\%$ compounded semi$-$annually. It provided him with payments of $$2,000$ at the end of every month for $6$ years. If the first withdrawal is to be made in $4$ years and $1$ month, how much did he pay for it$?$

$2)$ Melanie invested his savings in a bank at $3\mathrm{.}00\%$ compounded monthly. How much money did he invest to enable withdrawals of $$4,000$ at the beginning of every $6$ months from the investment for $6$ years, if the first withdrawal is to be made in $10$ years?

$3)$ Kristen set up a fund that would pay his family $$4,000$ at the beginning of every month, in perpetuity. What was the size of the investment in the fund if it was earning $3\mathrm{.}00\%$ compounded semi$-$annually?

$4)$ A college plans to set up an endowment fund that will provide a scholarship of $$5,000$ at the end of every quarter, in perpetuity. How much should the college invest in the fund, if the fund earns $3\mathrm{.}75\%$ compounded quarterly?

$5)$ If the market value of a telecommunications share is $$296\mathrm{.}15,$ calculate the year$-$end dividends that it should be able to pay in perpetuity if money is worth $4\mathrm{.}75\%$ compounded semi$-$annually.

$6)$ The Patchmans have decided to invest in a college fund for their young son. They invested $$30,000$ in a deferred annuity that will pay their son at the beginning of every month for $4$ years, while he goes to college. If the account earns $3\mathrm{.}00\%$ compounded monthly and the annuity payments are deferred for $12$ years, what will be the size of the monthly payments?