You would like to have $$$$50000$ in $16$ years. To accumulate this$$ amount, you plan to deposit an equal sum in the bank each year that will earn $6$ percent interest compounded annually. Your first payment will be made at the end of the year.

a$.$How much must you deposit annually to accumulate this$$ amount?

b$.$If you decide to make a large$$ lump$-$sum deposit today instead of the annual$$ deposits, how large should this$$ lump$-$sum deposit$$ be$?($Assume you can earn $6$ percent on this$$ deposit.$)$

c$.$At the end of five$$ years, you will receive $$$$10000$ and deposit this in the bank toward your goal of $$$$50000$ at the end of year $16.$ In addition to the$$ lump$-$sum deposit, how much must you deposit in equal annual$$ amounts, beginning in year $1$ to reach your$$ goal? $($Again$,$ assume you can earn $6$ percent on your$$ deposits.$)$