Suppose you start saving for retirement when you are $40$ years old. You

invest $$4,000$ the first year and increase this amount by $3\%$ each year to

match inflation. How much money will you have saved after $20$ years if

the interest rate earned on the retirement account is $9\%$ per year?

$($a$)$ You will have saved

$($Round to

the nearest dollar.$)$

$($b$)$ On the other hand, you will have saved

if the

rate of increase of the investment deposits was $9\%$ instead of the

original $3\%.$