What is the ruling on purchasing a car in installments from the outset, and what is the ruling on taking a loan to pay the remaining cost of it when there is an increase in the amount paid back to the bank?

If the aforementioned loan is an interest-based (riba) loan, then it is not permissible to take it, even if it leads to saving money, because borrowing with interest is one of the major sins and is not permissible except out of a recognized necessity. As for purchasing a car in installments, there is no harm in it if the method of purchase and the terms of the contract are Islamically sound, even if the price differs between immediate payment and installment.