Join the growing community of curious minds on IDNLearn.com and get the answers you need. Get accurate answers to your questions from our community of experts who are always ready to provide timely and relevant solutions.
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
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Your search for answers ends at IDNLearn.com. Thanks for visiting, and we look forward to helping you again soon.
