Question

A certain ball, when dropped from a height rebounds to one half of the original height.Suppose thos ball is dropped from a point 8ft above the ground, how far has it traveled, counting up and down distance only, when it hits the ground for the eight time?

Answer 1

distance of 1st bounce --- 8
distance of 2nd bounce --- 2(4) = 8
distance of 3rd bounce --- 2(2) = 4
distance of 4th bounce --- 2(1) = 2
distance of 5th bounce --- 2(1/2) = 1
distance of 6th bounce --- 2(1/4) = 1/2
distance of 7th bounce --- 2(1/8) = 1/4
distance of 8th bounce --- 2(1/16)= 1/8

Since there are only 8 terms, we can just add them up
I got a sum of 191/8

If there had been more terms, notice that , not counting the first term, we have a GP with a = 8 and r = 1/2
sum = 8 + sum(7)
= 8 + 8(1 - (1/2)^7)/(1- 1/2)
= 8 + 8(127/128) / (1/2)
= 8 + 8(127/128)(2)
= 191/8 , just as above