$($c$)$ You would like to have $50$ million cash in $15$ years time. To accumulate this amount you plan to deposit each year an equal sum in the bank, which will earn $7$ percent interest compounded annually. Your first payment will be made at the end of the year. There are no bank charges.

$($i$)$ How much must you deposit annually to accumulate to this amount?

$($ii$)$ If you decide to make a large lump$-$sum deposit $($same bank, same terms$)$ today instead of the annual deposits, how large should this lump$-$sum deposit be$?$