The Australian Open tennis tournament consists of 7 rounds of matches: Rounds 1, 2, 3, 4, the quarter-finals, the semi-finals and the finals. Half of the players (the losers of their matches) are removed after each round, so that there are only 2 players remaining in the final.(a) Starting at the final and working back through each round, write the number of players playing in each round as a number in index form.NOTE: INDICES ANSWERS ARE TO BE TYPED AS FOLLOWS: 5^3 for (b) Evaluate how many players are there at the start of the championship
Question
The Australian Open tennis tournament consists of 7 rounds of matches: Rounds 1, 2, 3, 4, the quarter-finals, the semi-finals and the finals. Half of the players (the losers of their matches) are removed after each round, so that there are only 2 players remaining in the final.(a) Starting at the final and working back through each round, write the number of players playing in each round as a number in index form.NOTE: INDICES ANSWERS ARE TO BE TYPED AS FOLLOWS: 5^3 for (b) Evaluate how many players are there at the start of the championship
Solution
(a) Starting from the final round, we have 2 players. This can be written as 2^1.
In the semi-finals, we have double the number of players, which is 4. This can be written as 2^2.
In the quarter-finals, we have double the number of players from the semi-finals, which is 8. This can be written as 2^3.
In the 4th round, we have double the number of players from the quarter-finals, which is 16. This can be written as 2^4.
In the 3rd round, we have double the number of players from the 4th round, which is 32. This can be written as 2^5.
In the 2nd round, we have double the number of players from the 3rd round, which is 64. This can be written as 2^6.
In the 1st round, we have double the number of players from the 2nd round, which is 128. This can be written as 2^7.
(b) To evaluate how many players are there at the start of the championship, we look at the 1st round. There are 2^7 players, which equals 128 players.
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