Can you figure out the number of players in the tournament?

It is getting warm and school is almost out, so I thought I would give a simple problem that is all insight and very little calculation.

An Internet tournament is being held for the board game Monopoly. If you do not like Monopoly, substitute any other game that can have four players. There will be four players per game. The tournament is single elimination. In each game, only the winner gets to play more games. There will be a total of 1365 games played. How many people are needed to register for the tournament? One simple multiplication and an even simpler addition will give the answer.

When I give the answer, I will show how the idea behind it can be used to give a completely intuitive derivation of the formula for geometric series, where each term has meaning. If you see where I am headed with this, feel free to jump in.