please help! math 20

Question

a ball kicked 32m up into the air. the ball falls to the graound, rebound to half of its original height and then falls to ground again. this pattern of rebounding to half of the height from the previos bounce continue. if this continue forever determine the total distance the ball travel.

Answer 1

Ball is kicked into the air = 32 m
It falls back 32 m
Total distance at the first bounce = 2×32 = 64
The ball falls to the ground, rebound to half of its original height.
Hence a= 64 and r = 1/2
The terms are 64 , 64×1/2, 64×(1/2)² , 64×(1/2)³.......

Sum of the geometric series formula:S_n =a(1−rⁿ)/(1−r)

Hence distance traveled in 'n' bounces =64(1−(½)ⁿ)/(1−½)
                                                       =64×2(1−(½)ⁿ)

                                                        =128(1−(½)ⁿ)

So total distance the ball travel is 128(1−(½)ⁿ)