Your local gift shop starts a membership program for its loyal customers. Customers who have $5-$

year membership card will get $15\%$ discount on every purchase and the store owner believes that a

loyal customer spends $$175($before discount$)$ on average in the store every year. The market

interest rate is $8\%.$

a$)$ If the store owner$$s expectations are correct, how much can she charge for the $5-$year

membership card? $($Note: Assume the customers make all their purchases at the end of the

year.$)$

b$)$ The owner also introduces another type of membership program: A perpetual membership.

Customers who own this type of membership card get $15\%$ discount on every purchase but they

benefit from this discount forever. The owner believes that those who want this card spend

$$200($before discount$)$ annually. If this is the case, what will be the price difference between the

$5-$year membership card and the perpetual membership card?