Answer
The first six terms of the geometric sequence are 8, 6, 4.5, 3.4, 2.5 and 1.9.
The ball is "good" that rebounds to a height of 2.5 feet after the fifth bounce.
Work Step by Step
For the geometric sequence, $a_{1} = 8$ and $r = \frac{3}{4}$, the general term $a_{n}=8(\frac{3}{4})^{n-1}$ where n is the number of rebound.
The first six terms of the geometric sequence are respectively
$a_{1}=8(\frac{3}{4})^{1-1} = 8$
$a_{2}=8(\frac{3}{4})^{2-1} = 6$
$a_{3}=8(\frac{3}{4})^{3-1} = 4.5$
$a_{4}=8(\frac{3}{4})^{4-1} = 3.4$ (round to the nearest tenth)
$a_{5}=8(\frac{3}{4})^{5-1} = 2.5$ (round to the nearest tenth)
$a_{6}=8(\frac{3}{4})^{6-1} = 1.9$ (round to the nearest tenth)
The first six terms of the geometric sequence are 8, 6, 4.5, 3.4, 2.5 and 1.9.
Since $a_{5} = 8(\frac{3}{4})^{5-1} = 2.53125$ which is $\gt$ 2.5, therefore, the ball is "good".
