Question

A ball is dropped from a height of 16 feet. Each time the ball dropped h feet, it rebounds 0.81h feet. Find the total vertical distanced travelled by the ball.

Answer 1

This is an example of infinite geometric sequence.
Formula for sum of infinite geom sequence:
S = a1 / (1 - r)

Total distance traveled:
d = (distance ball dropped) + (1st bounce up) + (1st bounce down) + (2nd bounce up) + (2nd bounce down) + ...
d = 16 + 16(0.8) + 16(0.8) + 16(0.8)(0.8) + 16(0.8)(0.8) + ...
d = 16 + 32(0.8) + 32(0.8)(0.8) + ...
d = 16 + (0.8)(32 + 32(0.8) + 32(0.8)(0.8) + ...)

we can see here that the infinite geometric sequence applies on the terms after 16, specifically on the (32 + 32(0.8) + 32(0.8)(0.8) + ...). Thus, a1 = 32 and r = 0.8. Using the formula,
d = 16 + (0.8)(32 / (1 - 0.8))
d = 144 ft.

hope this helps? `u`

Answer 2

Sorry. I see that r = 0.81 according to the problem, not 0.8.
Well substituting again,
d = 16 + (0.81)(32 / (1 - 0.81))
d = 152.4 ft.