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Sagot :
Using a geometric sequence, it is found that the rule for the number of matches played in the nth round is given by:
[tex]a_n = 64\left(\frac{1}{2}\right)^n[/tex]
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:
[tex]a_n = a_1q^{n-1}[/tex]
In which [tex]a_1[/tex] 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 [tex]a_1 = 64[/tex].
- In each successive round, the number of matches played decreases by one half, hence the common ratio is [tex]q = \frac{1}{2}[/tex].
Thus, the rule is:
[tex]a_n = 64\left(\frac{1}{2}\right)^n[/tex]
The last round is the final, in which 1 game is played, hence:
[tex]1 = 64\left(\frac{1}{2}\right)^n[/tex]
[tex]\left(\frac{1}{2}\right)^n = \frac{1}{64}[/tex]
[tex]\left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^6[/tex]
[tex]n = 6[/tex]
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 https://brainly.com/question/11847927
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