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samuelerlikhman

28.11.2021

Mathematics

teams

Step-by-step explanation:

Let's do it by parts.

Let's consider a fact: there has to be only one winner. After six games:

So, $a_{6}=1$

Since it's a decreasing Geometric Sequence (we have to lead to one winner), and each match two teams play, the beaten one leaves the championship.

$r=(\frac{1}{2})$

$a_{1}=?$

a) Explicit Rule (Formula):

$a_{6}=1,a_{1}=?r=\frac{1}{2}\\1=a_{1}\left ( \frac{1}{2} \right )^{6-1}\\1=a_{1}\left ( \frac{1}{2} \right )^{5}\\1=\frac{a_{1}}{32}\\a_{1}=32$

That's to say the tournament started with 32 teams

$\{32,16,8,4,2,1\}$

b) Recursive Rule

If we have the first term of the Geometric Sequence we can recursively find the subsequent terms. The Recursive Rule is useful when we have the prior term and we need the following one. Also, to derive the explicit rule.

$a_{1}=32\\a_{n}=\left ( \frac{1}{2} \right )a_{n+1}\\a_{2}=\left ( \frac{1}{2} \right )32=16\\a_{3}=\left ( \frac{1}{2} \right )16=8\\a_{4}=\left ( \frac{1}{2} \right )8=4\\a_{5}=\left ( \frac{1}{2} \right )4=2\\a_{6}=\left ( \frac{1}{2} \right )2=1\\$

$\{32,16,8,4,2,1\}$

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