You are trying to decide how much to save for retirement. Assume you plan to save $$6,500$ per year with the first investment made one year from now. You think you can earn $6\%$ per year on your investments and you plan to retire in $27$ years, immediately after making your last $$6,500$ investment.

a$.$ How much will you have in your retirement account on the day you retire?

b$.$ If$,$ instead of investing $$6,500$ per year, you wanted to make one lump$-$sum investment today for your retirement that will result in the same retirement saving, how much would that lump sum need to be$?$

c$.$ If you hope to live for $16$ years in retirement, how much can you withdraw every year in retirement $($starting one year after retirement$)$ so that you will just exhaust your savings with the $16$ th withdrawal $($assume your savings will continue to earn $6\%$ in retirement$)?$

d$.$ If$,$ instead, you decide to withdraw $$40,000$ per year in retirement $($again with the first withdrawal one year after retiring$),$ how many years will it take until you exhaust your savings?

e$.$ Assuming the most you can afford to save is $$1,300$ per year, but you want to retire with $$1,000,000\mathrm{.}00$ in your investment account, how high of a return do you need to earn on your investments?