$(15$ points$)$ You would like to have $$10,000,000$ when you retire in $45$ years. You believe

that any money you invest today can earn $12\%$ per year.

a$.$ How much of a single investment would you have to make today to reach your goal?

b$.$ If you set up a retirement account and plan to save a constant amount each month from

your paychecks, how much you will need to contribute each month to reach your goal?

c$.$ Assume you have reached your retirement goal with $$10,000,000$ and are retired now. At

this time, you want to take an annuity that pays over the next $20$ years. You have an

annual interest rate of $3\%.$ What would be the monthly annuity payment?