This section contains problems $8$ to $10$ out of $10$ problems. Each problem is worth $7$ points. Please show your work for each problem for credit. Ignore transaction costs and taxes for the problems. Please fit all your work for the problems on one side of the paper, write your name on the sheet, make a digital file of the sheet, and upload the file.

$8.$ You want to save money for your retirement. You plan to retire in $35$ years.

a$)$ If you deposit $$2,500$ at the end of every month for the next $35$ years into a retirement account that provides $9$ percent annual return compounded monthly, how much money will you have in the account in $35$ years? Assume that the first deposit occurs at the end of the first month and all deposits occur at the end of the month.

b$)$ One month after making you last $$2,500$ deposit, you will make an equal withdrawal from the account monthly for the next $25$ years. Assume that you earn $6$ percent annual return compounded monthly during this period, how much can you withdraw each month so that the balance in the account is zero when you make the last withdrawal?

$9.$ You make the following deposits for the next five years into an investment account. All deposits are made at the end of the year and the first deposit occurs one year from now. No more deposits are made after year $5.$ You will leave all the money in the account until year $35$ from now. If you earn $8$ percent annual return for the first five years and $6$ percent annual return for all subsequent years, how much will you have in the account at the end of year $35?$

$\backslash$table$[[$Year$,$Deposit$],[1,$$$120,000$