Numbers: How to find all relevant formula's variables

Numbers does not have a "solver" capability. I think Excel can do it with with its Solver add-in. I tried it on this particular equation and it worked fine, getting an answer in a second or so. Excel might be your best solution unless someone has written a solver capability in AppleScript for Numbers. This same type of problem has come up occasionally but I don't recall anyone tackling it.

One other solution is to use Desmos.

https://www.desmos.com/calculator

Here is an example (not your function):

"y" is your "A"

"x" is your "n"

The first equation is the formula you are evaluating. The second is a straight line equal to the answer (y) for which you want to find x.

If you click on the intersection of the two on the graph, it gives you the coordinates. The answer in this case is -16.201.

With the caveat that I am not a mathematician and this method I am about to present may not actually work, below is a possible solution. It requires wisely choosing two starting points for "n". For the function you posted you don't have to be all that wise to choose the starting points but you do if the function has local minimums and maximums. If one point is on one side of a minimum/maximum and the other is on the other side, you might find a different answer or no answer. It will find only one answer for a given set of starting points, even if there are multiple answers.

A2 has an "n" value so we can calculate a value in B2. You could just type a number into cell B2 but I wanted to see if the method got the same "n" as I started with.

C2 and D2 are the other constants, typed in.

B2 =C2×(D2×(1+D2)^A2)÷((1+D2)^A2−1)

fill down to complete the column.

C3 = C2

D3 = D2

Fill down with both to complete the columns

A3 and A4 are two starting points you have to choose wisely.

A5 =IF($A3=$A4,$A4,FORECAST($B$2,$A3:$A4,B3:B4))

fill down from A5

For choosing the initial points, it might be helpful to make a chart of the function over a range of input values so you can see where the minimums and maximums are.

It appears you want to be able to choose which variable is missing and have the table calculate that missing variable. Then you want to be able to easily choose a different one and have it calculate that one instead. You can do that pretty easily with Excel's Solver.

The method I am using is pretty simple and blows up easily with discontinuous functions and those that go to infinity (like some of your examples when solving for n) unless you choose the initial points well. I suppose with a smarter algorithm that will take smaller jumps it might be more stable but it is still pretty simple. I have an idea that would make it easy to change which variable to solve for, just a couple of clicks on checkboxes, but it doesn't solve your discontinuous functions all that well so probably it isn't worth the effort.

Actually, it can probably be solved in a table with sufficient rows but it may not be practical. Offhand I don't know how to do it. It is a minimization problem.

If I understand, you have A, S, and i and want to find n?