Is it permissible to buy a car offered for sale for $3500 cash, after it was sold to the same buyer for $5500 in installments, and is the price difference permissible?

The transaction mentioned is Bay' al-'Inah (a type of buy-back sale), which is a matter of scholarly disagreement. If there is no collusion between the two parties, then the matter is closer to permissibility, and its prohibition would be a preventative measure (sadd al-dhari'ah) rather than due to the presence of usury (riba). Sheikh al-Islam Ibn Taymiyyah holds that if 'Inah involves collusion, both sales are invalidated. If there is no collusion, the second sale is invalidated as a preventative measure. If the buyer's return to purchase the car again is after some time and the purpose was not to obtain an increase in price, then the more apparent view is that it is not forbidden.