How much money is left after the first day?

Let x = the money he started with, and let's say a = the money remaining after the first day.

So, after Day 1:

That's the algebraic way of saying, he paid a buck to get through the door of the casino, then lost half of what he had. On the way out, he paid a buck to leave.

The other way to express that formula, by the way, is this:

What about after Day 2?

Well, the above formula is the amount left after Day 1, right? So, to figure out what's left after the second day, instead of x, we're going to substitute the amount left after the first day, or

So, the amount left after the second day is the amount left after the first day, or
minus a dollar, divided by two, and then minus another dollar, or

And so on, if you really wanted to solve this algebraically.

Personally, we found it much easier to just work backwards, and guess.

Pretty elegant, though, eh? We sure thought so.