I made up the following situation/problem but I'd like to know if they have an specific name so I can look for more examples.
I buy a car for \$20,000 and during the first month it requires an amount of gas worth \$100. This cost doubles every month because the car is leaking. All the cars in existence cost the same and suffer the same problem. Every how many months is it worth to buy a new car rather than keep paying the amount of gas?
Also, I'd like to check if my approach is correct:
One solution that it occurs to me is that at the end of the ninth month, I would be paying \$25,600 in gas so that month would be an obvious moment to buy a new car, "reset" the cost of gas so that month I pay \$20,100 instead.
In terms of the function that gives the cost based on the month:
$$ \begin{eqnarray} f(1) = 20,100 \\ f(n) = 100\cdot2^{n-1} \end{eqnarray} $$
Where I have solved for $ f(n) > 20,100 $ which gives me $ n\approx8.65 $, i.e. the ninth month I'd be paying more than the first month already.
Do this word problem have a specific name? Also, is my approach correct?
I would simply call it calculus of interest (with compound interests), but instead of calculation with money, you calculate with different items here.
From a business perspective, you could also view this problem as a cost-benefit calculation.