Question $1.$ You are a loan manager at Bank. Your client, Jordan, is setting up a new small business and

takes out a loan from your bank for $$100,000$ on October $1,2023.$ You offer Jordan the following repayment scheme:

Payments are to be made at the beginning of each month, starting October $1,2024.$

The loan accrues interest at a reduced APR of $7\%,$ compounding at the end of each month in the first year.

The loan then accrues interest at an APR of $10\%,$ compounding at the end of each month starting in the second

year $($October $2024)$ and onward.

$($a$)$ Jordan wants his last payment to be on September $1,2039.$ Calculate Jordan's monthly payment. $($Round your

answer to the nearest cent.$)$

$($b$)$ After five years, Jordan's business is very successful, and he can afford to pay more each month. Suppose he

doubles his monthly payments starting October $1,2029.$ When would Jordan's last payment be$?$