Question

There are 128 teams entered in a basketball tournament. Half of the teams are eliminated each round. How many teams are left after 4 rounds?

Answer 1

We have 128 teams.

In each round, half of the teams remain, so if T is the number of teams and n is the number of rounds, we have:

`T(n)=0.5\cdot T(n-1)`

and we start with n=0 so T(0)=128.

Then, we can use this recursive expression to find T after four rounds (n=4), but it is better if we derive an explicit formula for T in function of n (and not in function of the past terms of the sequence):

`\begin{gathered} T(1)=0.5\cdot T(0) \\ T(2)=0.5\cdot T(1)=0.5\cdot0.5\cdot T(0)=0.5^2\cdot T(0) \\ T(n)=0.5^n\cdot T(0)=0.5^n\cdot128 \end{gathered}`

With the last expression we can calculate T for any value of n. Then, for n=4, we have:

`T(4)=0.5^4\cdot128=((1)/(2))^4\cdot128=(1)/(2^4)\cdot128=(1)/(16)\cdot128=8`

NOTE: in this case, its better to express the factor 0.5 as a fraction instead of a decimal.

Answer: after 4 rounds there will be 8 teams left.