Respuesta :
Given:
The height from whihc the ball is dropped, h=20 feet.
The height attained by the ball at each bounce can be written as a geometeric series.
Let a=20 feet be the first term of the series.
Since the ball bounces 1/4 as high as the preceding one, the common ratio of the sequence is,
[tex]r=\frac{1}{4}[/tex]The sum of n terms in a geometric sequence is,
[tex]S_n=\frac{a(1-r^n)}{1-r}[/tex]The total height traveled by the 8th bounce is given by the sum of 8 terms in a geometric series starting from a=20 ft.
The sum of the terms in a GP with a=20, r=1/4 and n=8 is,
[tex]S_8=\frac{20(1-(\frac{1}{4})^8)}{(1-\frac{1}{4})}=26.66[/tex]Now, the total height traveled by the 8th bounce is,
[tex]H=2\times S_8-a=2\times26.66-20=33.32\text{ ft}[/tex]Hence, the total height the ball would have traveled by the 8th bounce is 33.32 ft.
