Question $2.$ On January $1,2024,$ you are setting up a perpetuity where you can withdraw $$20000$ at the end of

every quarter where the first withdrawal occurs on March $31,2024.$ To do this, you make an initial deposit of A into

an account where interest is compounded at the end of each month at an APR of $r.$

$($a$)$ Determine $A$ in terms of $r.$

$($b$)$ The terms of the perpetuity change after $10$ years, so that interest is to be compounded continuously at an APR

of $r$ instead. Determine which of the two following statements is true, and justify your answer.

Statement A: You need to deposit a lump sum into the account at the $10-$year mark so that you can still

receive the quarterly payments of $$20000.$

Statement B: The revised terms of the perpetuity allow you to withdraw more money every quarter

without adding any more money into the account.