# What do you think of this derivation of the formula for geometric series?

Asked by LostInParadise (32043) February 11th, 2023

The geometric series formula is 1 + x + x^^2…+x^^n = (x^^n+1 – 1)/(x-1). It can be fairly easily derived algebraically, but here is a completely non-algebraic derivation for the case where x is a positive whole number.

Suppose there is a tournament for the game of Monopoly, or whatever multi-player game you prefer. We start with 81 players and there will be 3 players per game. In the first round there are 27 games, and only the winners in each round go on to the next round.

The total number of games is the geometric series 27+9+3+1. Here is another way of getting the total number of games. At the end of the tournament there is a single winner, meaning that 80 players were eliminated. Since two players are eliminated in each game, the total number of games is 80/2 = 40.

This is easily generalized for x players per game and x^^(n+1) players in the first round. The total number of games is the geometric series 1+x+...+x^^n and is also obtained by dividing the x^^(n+1) – 1 players eliminated by the x-1 players eliminated per game.

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I can only admire it.

To help me separate the principle from the specifics, can you say what happens if there are four players per game?

Jeruba (55943)

Let n = 3. We start with 4^^(n+1) = 4^^4 = 256 players. The number of games in the first round is 64. The total number of games in all the rounds is 64+16+4+1=85.

The other way of looking at it is to say that we need to eliminate 255 players and, since we eliminate 3 players per game, the total number of games is 255/3=85.