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

`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 the of the geometric sequence and r be the common ration.

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

`a_n = ar^(n-1)`

`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)\\\\\Rightarrow r^2 = (1)/(4)\\\Rightarrow r = (1)/(2)`

the first term can be calculated as:

`16=a((1)/(2))^3\\\\a = 16* 6\\a = 128`

Thus, the required geometric sequence is

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