A ball is dropped from a height of 3 ft The elasticity of th

A ball is dropped from a height of 3 ft. The elasticity of the ball is such that it always bounces up one-third the distance it has fallen.

(a) Find the total distance the ball has traveled at the instant it hits the ground the fifth time. (Enter an exact number.)

(b) Find a formula for the total distance the ball has traveled at the instant it hits the ground the nth time. an =

Solution

The height of the ball follows the following sequence

3,1,1/3, 1/9 ......

That sequence follows geometric progression

Where first term is 3

common ratio=1/3

And sum of n terms of GP is

S_n=a(1-rⁿ)/(1-r)

     = 3(1-1/3⁵)/1-1/3

     = 3*242/243/ 2/3 =242/81/2/3= 242*3/2*81     = 121/27

b. a_n=a(1-rⁿ)/(1-r)