Chapter 6.5, Problem 33E

To determine

To find:The number of teams which remain after the third, fourth and fifth rounds.

Answer to Problem 33E

After third round 16 teams remain, after fourth round 8 teams remain, after fifth round 4 teams remain.

Explanation of Solution

Given information:A badminton tournament begins with 128 teams. After the first round, 64 teams remain. After the second round, 32 teams remain.

Calculation:Numberof teams at the beginning $=128$

Number of teams after 1^st round $=128\times \frac{1}{2}=64$

Number of teams after 2^nd round $=64\times \frac{1}{2}=32$

It is observed the number of team form a sequence in which each term is obtained by multiplying a constant $\frac{1}{2}$ with the preceding term.

Thus, $128,64,\mathrm{32...}$ is a geometric sequence with common ratio $\frac{1}{2}$

Then, number of teams after 3^rd round $=32\times \frac{1}{2}=16$

Number of teams after 4^th round $=16\times \frac{1}{2}=8$

Number of teams after 5^th round $=8\times \frac{1}{2}=4$

Thus, after third round 16 teams remain, after fourth round 8 teams remain, after fifth round 4 teams remain.