A single-elimination basketball tournament starts with 64 teams. The teams compete in paring until there is 1 winner. So, to find the number of rounds that must take place until the championship game a recursive formula is used i.e. [tex]a_1=64 , a_n=a_{(n-1)} (0.5)[/tex] is used. This is because a tournament between 64 teams represents a recursive geometric sequence.
Recursive function:
- A function that uses its previous term to calculate the next terms and forms a sequence of terms is called a recursive function
- This function is represented based on the arithmetic-geometric sequence as they have a common difference.
- For geometric sequence the recursive formula is [tex]a_n=a_{(n-1)} r[/tex] where [tex]a_n[/tex] is the nth term in the sequence and r is the common ratio of the terms
Calculating recursive formula for the given data:
Given that there are 64 teams in the basketball tournament
the teams compete in pairing until there is 1 winner.
At first, 64 teams start the tournament by competing in pairs. so, the total pairs in first-round = 32
Similarly,
in the second-round total pairs = 16
third-round total pairs = 8
fourth-round total pairs = 4
fifth-round = 2
sixth-round = 1
So, this forms a geometric sequence.
Hence, the recursive formula is given as,
[tex]a_1=64[/tex] and
[tex]r=\frac{1}{2}[/tex]
(r is the ratio of total pairs in the second round to the total pairs in the first round)
Therefore,
[tex]a_n=a_{(n-1)} r[/tex]
⇒ [tex]a_n=a_{(n-1)} (0.5)[/tex]
Thus, the recursive formula is used to find the number of rounds that must take place until the championship game is given [tex]a_n=a_{(n-1)}(0.5)[/tex].
To learn more about such recursive functions refer here:
https://brainly.com/question/983382
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