A wrestling tournament begins with 128 competitors. In the first round, each competitor has 1 match against another wrestler. The winner of each match moves on to the next round until there is a winner. Write a sequence in which the terms represent the number of players still in the tournament at the end of each round. Describe the sequence? How many rounds are in the tournament?

A wrestling tournament begins with 128 competitors. In every round competitor has 1 match against another wrestler and winner moves to the next round. We have to construct a sequence representing all rounds of this tournament.

In first round total matches played = 128/2 = 64

In second round number of matches played = 64/2 = 32

In third round matches played = 32/2 = 16

Similarly all the rounds were played.

Now we get the sequence as 64, 32, 16, 8, 4, 2, 1

This sequence is a geometric sequence having first term 64

Common factor = 32/64 = 1/2

Number of terms = 7

Explicit rule [tex]a_{n}= a(r)^{n-1}=64.(\frac{1}{2})^{n-1}[/tex]

Total seven rounds were played.

MrRoyal MrRoyal · Answer 2 · 2022-01-21T16:00:48+01:00

The sequence is a geometric sequence, and the number of rounds in the tournament is 7

From the question, we have the following parameters:

There are 128 competitors, initially
Half of the competitors progress, after each round.
There is only one winner

So, we have:

[tex]a =128[/tex]

[tex]r = \frac 12[/tex]

[tex]L =1[/tex]

The last term of a geometric sequence is calculated as:

[tex]L = ar^{n-1}[/tex]

So, we have:

[tex]1 = 128 \times (\frac 12)^{n-1}[/tex]

Divide both sides by 128

[tex]\frac{1}{128} = (\frac 12)^{n-1}[/tex]

Rewrite as:

[tex](\frac 12)^7 = (\frac 12)^{n-1}[/tex]

Cancel out the base

[tex]7 = n-1[/tex]

Rewrite as;

[tex]n -1 = 7[/tex]

This gives

[tex]Round = 7[/tex]

Hence, the number of rounds in the tournament is 7

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