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Part $1$

You are saving for retirement. To live$$ comfortably, you decide you will need to save $$$$2$ comma $000$ comma $000$ by the time you are age $67.$ Today is your $30$th $$birthday$,$ and you$$ decide, starting today and continuing on every birthday up to and including your $67$th $$birthday$,$ that you will put the same amount into a savings account. If the interest rate is $10\%,$ you set aside $$$$5$ comma $493\mathrm{.}85$ each year to make sure that you will have $$$$2$ comma $000$ comma $000$ in the account on your $67$th birthday. You realize that your plan has a flaw. Because your income will increase over your$$ lifetime, it would be more realistic to save less now and more later. Instead of putting the same amount aside each$$ year, you decide to let the amount that you set aside grow by $8\%$ per year. Under this$$ plan, how much will you put into the account$$ today? $($Recall that you are planning to make the first contribution to the account$$ today.$)$