A ball rebounds to one quarter from which it was dropped. approximate the total vertical distance the ball after being dropped from 3ft above the ground ,until it comes to rest

Answer 1

first bounce --- 3 ft
2nd bounce = 2(1/4)(3) = 6(1/4)
3rd bounce = 2(1/4)(3)(1/4) = (6)(1/4)^2
4th bounce = ... = (6)(1/4)^3
....

so we have
3 + 6(1/4) + 6(1/4)^2 + 6(1/4)^3 + ... to infinity
= 3 + 6(1/4) (1 + 1/4 + (1/4)^2 + ...)

from the 2nd term on, we have a geometric series
with a = 1 , r = 1/4

sum of all terms
= 3 + (6/4) S∞
= 3 + (3/2)(1/(1-1/4)
= 3 + (3/2)(4/3)
= 5