Question

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?

Answer 1

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.

What is a geometric sequence?

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