The profit up to 9800 is $49000.

Beyond that now, the profit drops by $0.0003 for each additional bottle.

So, for first bottle (after 9800), the profit is $4.9997

For the second (after 9800), the profit is $4.9994

etc.

The profit of the nth bottle (after 9800) is thus given by

The sum of profit for the bottles beyond 9800 is then given by:

So, the total profit becomes:

Now, to find the maximum, we find the derivative and set to zero.

Well, since it's not an integer let's test both the lower and upper integers.

When n = 16666, P = $90664.1667

When n = 16667, P = $90664.1666

Uh... well... the difference is so small that it is found to a hundredth of a cent...

But you got your answer. I derived the profit formulae from the arithmetric progressions formulae and modified them to suit the situation.