Question

A ball has bounce coefficient 0 < r < 1 if when it is dropped from height h, it bounces back to a height of rh. Suppose that such a ball is dropped from an initial height a and subsequently bounces infinitely may times. Find the total up-and-down distance in all its bouncing.

Answer 1

h[1 + 2r + 2r^2 + 2r^3 + ...]

= 2h(1 +r +r^2 + r^3 + ..) -h
= 2h [1/(1-r)] -h

Answer 2

distance= h+2hr+2hr^2+2hr^3...

distance= 2h+2hr+ 2hr^2+... -h
Isn't that an geometric series?
http://en.wikipedia.org/wiki/Geometric_progression

sum= a/(1-r) -h= 2h/(1-r) -h
check my thinking