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:
[tex]T(n)=0.5\cdot T(n-1)[/tex]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):
[tex]\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}[/tex]With the last expression we can calculate T for any value of n. Then, for n=4, we have:
[tex]T(4)=0.5^4\cdot128=(\frac{1}{2})^4\cdot128=\frac{1}{2^4}\cdot128=\frac{1}{16}\cdot128=8[/tex]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.
