Question

The numbers of teams remaining in each round of a single-elimination tennis tournament represent a geometric sequence where an is the number of teams competing and n is the round. There are 16 teams remaining in round 4 and 4 teams in round 6.

The explicit rule for the geometric sequence is:

Answer 1

The geometric series is given by:

`a_n = 128\bigg((1)/(2)\bigg)^(n-1)`

Explanation:

We are given the following in the question:

The numbers of teams remaining in each round follows a geometric sequence.

Let a be the first term and r be the common ratio.

The
`n^(th)` term of a geometric sequence is given by:

`a_n = ar^(n-1)`

There are 16 teams remaining in round 4 and 4 teams in round 6.

Thus, we can write:

`a_4 = 16 = ar^3\\a_6 = 4 = ar^5`

Dividing the two equations, we get,

`(16)/(4)=(ar^3)/(ar^5)\\\\4=(1)/(r^2)\\r^2=(1)/(4)\\\\r = (1)/(2)`

Putting value of r in he equation we get:

`16=a((1)/(2))^2\\\\a = 16* 2^3\\a = 128`

Thus, the geometric sequence can be written as:

`a_n = 128\bigg((1)/(2)\bigg)^(n-1)`