Ruth is scheduling a soccer tournament in which 64 teams will participate. After each round of soccer games, half the teams advance to the next round of the tournament. After all the rounds are played, how many total games are in the tournament?

Is it 48?

32 + 16 + 8 + 4 + 2 + 1 = ?

Thank you!!!!

No, the correct answer is 63.

To understand how this is calculated, we can break it down step by step:

In the first round, there are 64 teams competing. Since half of them advance to the next round, this means that 32 teams will move forward.

In the second round, there are 32 teams remaining. Again, half of them advance, leaving 16 teams.

This pattern continues until we reach the final round, where only one team will be left.

To calculate the total number of games, we need to sum up the number of games in each round. In each round, half the number of teams participate in a game (because one team will be eliminated). Since each game involves two teams, we divide the number of teams by 2 to get the number of games.

In the first round, there are 64 teams. Dividing by 2 gives us 32 games.
In the second round, there are 32 teams remaining, which means 16 games.
In the third round, there are 16 teams remaining, which means 8 games.
And so on, until we reach the final round with only 1 game.

To calculate the total number of games, we sum up the individual counts: 32 + 16 + 8 + ... + 1.

To simplify this, we can use the formula for the sum of a geometric series:

Sn = a(1 - r^n) / (1 - r)

where Sn is the sum, a is the first term, r is the common ratio, and n is the number of terms.

In this case, a = 32 (the first term), r = 1/2 (since half the teams advance), and n = 6 (since there are 6 rounds).

Using the formula, we have:

Sn = 32(1 - (1/2)^6) / (1 - 1/2)
= 32(1 - 1/64) / (1/2)
= 32(63/64) / (1/2)
= (32 * 63 * 2) / 64
= 2016 / 64
= 63

Therefore, the total number of games in the tournament is 63.