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Part $1$

You are trying to decide how much to save for retirement. Assume you plan to save $$$$6$ comma $500$ per year with the first investment made one year from now. You think you can earn $10\%$ per year on your investments and you plan to retire in$42$years$,$ immediately after making your last $$$$6$ comma $500$ investment.

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

b$.$ If$,$ instead of investing $$$$6$ comma $500$ per$$ 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 $10\%$ in$$ retirement$)?$

d$.$ If$,$ instead, you decide to withdraw $$$$400$ comma $000$ per year in retirement$($again with the first withdrawal one year after$$ retiring$),$ how many years will it take until you exhaust your$$ savings?

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