Solution:
Let H represent the height.
Given that the initial height from which the rubber ball is dropped is 800 centimeters, this implies that
[tex]H_1=800[/tex]It is noticed that after each bounce, it reached 85% of its previous height. This implies that
[tex]H_2=\frac{85}{100}\times H_1=0.85H_1[/tex]Similarly,
[tex]H_3=\frac{85}{100}\times H_2=0.85H_2[/tex]Thus, since each height is attained by a common ratio, using the geometric sequence formula expressed as
[tex]\begin{gathered} T_n=ar^{n-1} \\ where \\ T_n\implies H_n \\ a\implies H_1 \\ r\text{ is the common ratio between consecutive terms eevaluated as} \\ \frac{H_2}{H_1}=\frac{0.85H_1}{H_1}=0.85 \end{gathered}[/tex]Thus, substituting these parameters into the geometric sequence formula, we have
[tex]\begin{gathered} H_n=H_1(0.85)^{n-1} \\ where \\ H_1=800 \\ \Rightarrow H_n=800(0.85)^{n-1} \end{gathered}[/tex]Hence, the equation model for the height H, for n bounces is
[tex]\begin{gathered} \begin{equation*} H_n=800(0.85)^{n-1} \end{equation*} \\ where \\ n=1,2,3,... \end{gathered}[/tex]The third option is the correct answer.
