$1.$ You are planning on how much you need to save for retirement. You expect to live for $30$ years in retirement and would like to spend $$100,000($in real terms$)$ per year, while leaving a $$1$ million bequest to Georgetown when you die at the end of the $30$th year. You are $35$ years away from retirement. How much do you need to save at the end of each year if you earn $5\%$ real $($after$-$inflation$)$ during your working years and $3\%$ during your retirement years? Solve the problem in two different ways: