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mina [271]
2 years ago
12

The men's U.S. Open tennis tournament is held annually in Flushing Meadow in New York City. In the first round of the tournament

, 64 matches are played. In each successive round, the number of matches played decreases by one half.
Find a rule for the number of matches played in the nth round. For what values of n does your rule make sense?
Mathematics
1 answer:
Dmitrij [34]2 years ago
7 0

Using a geometric sequence, it is found that the rule for the number of matches played in the nth round is given by:

a_n = 64\left(\frac{1}{2}\right)^n

The rule makes sense for values of n of at most 6, as in the last round, which is the 6th and final round, 1 game is played.

<h3>What is a geometric sequence?</h3>

A geometric sequence is a sequence in which the result of the division of consecutive terms is always the same, called common ratio q.

The nth term of a geometric sequence is given by:

a_n = a_1q^{n-1}

In which a_1 is the first term.

In this problem, we have that:

  • In the first round of the tournament, 64 matches are played, hence the first term is a_1 = 64.
  • In each successive round, the number of matches played decreases by one half, hence the common ratio is q = \frac{1}{2}.

Thus, the rule is:

a_n = 64\left(\frac{1}{2}\right)^n

The last round is the final, in which 1 game is played, hence:

1 = 64\left(\frac{1}{2}\right)^n

\left(\frac{1}{2}\right)^n = \frac{1}{64}

\left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^6

n = 6

Hence, the rule makes sense for values of n of at most 6, as in the last round, which is the 6th and final round, 1 game is played.

More can be learned about geometric sequences at brainly.com/question/11847927

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