1 unit of ingredient A and 2 units of ingredient B yields 6 units of ingredient A. This process can be repeated any number of times (i.e. those 6 new units of A can be combined with 12 units of B to make 36 units of A). What is a formula to determine how much of ingredient B I would need to reach a certain amount of ingredient A through repetitions of this process.
I suppose you start with 1 unit of ingredient A?
First thing I'd do is to write a table, and try to find how many units of A I'd get from the nth step, and how many units of B I'd use at that step, like this:
Step................................Ingredient A used..................Ingredient B used.........Ingredient A at end of step.............Total ingredient B used
1....................................................1.................................................. 2.................................................6............................................................ 2
2....................................................6...................................................12..............................................36........................................................2+6 [imath]\cdot[/imath] 2=14
...
n....................................................?.....................................................?..................................................?.............................................................?
See if you can find how much of ingredient A is obtained at the end of step n. It might be useful to also calculate step 3, step 4 (you'll have to get a rule, or a relation between the 4 "?" I put in the table.
Once you do that, you have to find a way to calculate how many steps you'd need to do to get x units of A. I guess we can try it out for, let's say, 1000 units of A, to get a clear idea about how to do it; you need to solve the first part to better understand what I mean here...
