Question

The numbers of teams remaining in each round of a single-elimination tennis tournament represent ageometric sequence where an is the number of teams competing and n is the round. There are 32 teamsremaining in round 4 and 8 teams in round 6.The explicit rule for the geometric sequence is

Answer 1

Given:

The numbers of teams remaining in each round of a single-elimination tennis tournament represent a geometric sequence

The general form of the geometric sequence is as follows:

`a_n=a_1\cdot r^{n-1^{}}`

Given the following conditions:

1) There are 32 teams remaining in round 4

So,

`\begin{gathered} 32=a_1\cdot r^(4-1) \\ 32=a_1\cdot r^3\rightarrow(1) \end{gathered}`

2) there are 8 teams in round 6

So,

`\begin{gathered} 8=a_1\cdot r^(6-1) \\ 8=a_1\cdot r^5\rightarrow(2) \end{gathered}`

So, we will solve the equations (1) and (2) to find a1, and (r)

Divide the equation (2) by (1) to eliminate a1, then solve for (r)

`\begin{gathered} (8)/(32)=(a_1\cdot r^5)/(a_1\cdot r^3) \\ \\ (1)/(4)=r^{^(5-3)}\rightarrow r^2=(1)/(4) \\ \\ r=\sqrt[]{(1)/(4)}=(1)/(2) \end{gathered}`

Substitute with (r) into equation (1) to find a1

`\begin{gathered} 32=a_1\cdot((1)/(2))^3 \\ 32=a_1\cdot(1)/(8) \\ a_1=32\cdot8=256 \end{gathered}`

So, the answer will be:

`a_n=256\cdot((1)/(2))^(n-1)`