To determine the number of players participating in the tennis tournament, we need to find the maximum number of matches that can be played.
In a knock-out tournament, each match eliminates one player. So, for every match played, one player is out of the tournament.
Let's assume the number of players participating in the tournament is 'n'.
In the first round, there are 'n/2' matches played, where 'n/2' is the total number of players divided by 2. After these matches, half of the players are eliminated.
In the second round, there are 'n/4' matches played, as half of the remaining players are eliminated again.
This pattern continues until there is only one match remaining, which is the final match.
So, the total number of matches played can be calculated using the formula:
Total Matches = (n/2) + (n/4) + (n/8) + ... + 1
We are given that the total number of matches is 63.
Hence, (n/2) + (n/4) + (n/8) + ... + 1 = 63
To simplify this equation, we can multiply both sides by 2:
n + (n/2) + (n/4) + ... + 2 = 126
Now, we can see that this is a geometric progression with a common ratio of 1/2.
Using the formula for the sum of a geometric progression, we have:
Sum = a(1 - r^n) / (1 - r)
where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
In this case, a = n, r = 1/2, and the sum is 126.
Plugging in these values, we get:
n(1 - (1/2)^n) / (1 - 1/2) = 126
2n - 2^n = 126
To solve this equation, we can observe that when n = 64, the equation holds true:
2(64) - 2^64 = 128 - 1 = 127
So, the number of players participating in the tournament is 64.
Therefore, the correct answer is option A) 64.