The men’s U.S open tennis court is held annually in Flushing Meadow New York City. In the first round of the tournament, 64 matches are played. In each successive round, the number of matches decreases by 1/2.
Make a rule for the number of matches played the nth round.
For what values of n does your rule make sense.
Find the number of matches played in the US open tennis tournament.
matches in nth round = 2^(7 - n)
total matches = 64 + 32 + ... + 1 = 2^7 - 1
To find a rule for the number of matches played in each round, we can start by looking at the pattern. In the first round, 64 matches are played. In each successive round, the number of matches decreases by 1/2. Let's break it down to understand the pattern:
First round: 64 matches
Second round: 64 * (1/2) = 32 matches
Third round: 32 * (1/2) = 16 matches
Fourth round: 16 * (1/2) = 8 matches
Fifth round: 8 * (1/2) = 4 matches
And so on...
From this pattern, we can observe that the number of matches played in each round is halved from the previous round. Therefore, we can formulate a rule for the number of matches played in the nth round:
Number of matches in the nth round = 64 * (1/2)^(n-1)
To determine which values of n the rule makes sense for, we need to consider if the expression inside the parenthesis (1/2)^(n-1) will always be a valid value. Since the exponent (n-1) is always at least 0 (for the first round), we don't have to worry about dividing by zero.
Therefore, the rule makes sense for all positive integer values of n, including 1, 2, 3, 4, and so on.
To find the total number of matches played in the US Open Tennis tournament, we need to sum up the number of matches played in each round:
Total number of matches = 64 + 32 + 16 + 8 + ... + (64 * (1/2)^(n-1))
Since the number of matches decreases by half in each round, this is a geometric series. The sum of a geometric series can be calculated using the formula:
Sum = a * (1 - r^n) / (1 - r)
In this case, the initial term (a) is 64, the common ratio (r) is 1/2, and the number of terms (n) is not specified. We need to find the sum of an infinite geometric series:
Total number of matches = 64 * (1 - (1/2)^n) / (1 - 1/2)
However, since the number of matches cannot be fractional, we need to determine the number of terms (n) that makes the expression inside the parenthesis become negligible. In other words, we need to find the smallest positive integer value of n for which (1/2)^n is less than or equal to 1/64.
To find this value, we can start by solving the inequality (1/2)^n ≤ 1/64:
(1/2)^n ≤ 1/64
Taking the logarithm of both sides (base 1/2):
log(base 1/2) [(1/2)^n] ≤ log(base 1/2) (1/64)
n ≤ log(base 1/2) (1/64)
Since log(base 1/2) (1/64) = -6, we have:
n ≤ -6
Therefore, the smallest positive integer value of n for which (1/2)^n is less than or equal to 1/64 is n = 7.
Hence, the total number of matches played in the US Open tennis tournament is:
Total number of matches = 64 * (1 - (1/2)^7) / (1 - 1/2) = 127.
To determine the rule for the number of matches played in each round, we start with the information given: 64 matches are played in the first round, and in each successive round, the number of matches decreases by 1/2.
Let's break down the problem step-by-step:
Step 1: Determine the number of matches played in each round.
- In the first round, there are 64 matches.
- In the second round, the number of matches decreases by 1/2, so there would be 64/2 = 32 matches.
- In the third round, again the number of matches decreases by 1/2, so there would be 32/2 = 16 matches.
- This pattern continues, with the number of matches halving in each round.
Step 2: Write a rule for the number of matches played in the nth round.
- From the pattern above, we can see that the number of matches in each round can be determined by the formula: 64/(2^(n-1)).
- In this formula, n represents the round number.
Step 3: Find the values of n for which the rule makes sense.
- The rule makes sense for any positive integer value of n since we can calculate the number of matches for each round based on the formula mentioned above.
Step 4: Find the total number of matches played in the US open tennis tournament.
- To find the total number of matches played in the tournament, we need to sum up the number of matches from all the rounds.
- The tournament consists of 64 matches in the first round, followed by 32 matches in the second round, 16 matches in the third round, and so on, until we reach the round where there is only one match.
- The sum of matches in the tournament would be:
64 + 32 + 16 + ... + 1.
- This is a geometric series with a common ratio of 1/2.
- We can use the sum of a geometric series formula to find the total number of matches, which is given by:
Total number of matches = 64 * (1 - (1/2)^n) / (1 - 1/2).
By following these steps, we can determine the rule for the number of matches played in each round, find the values of n for which the rule makes sense, and calculate the total number of matches played in the US open tennis tournament.