Aball is dropped from a height of 16 feet . each time it drops, it rebounds by half of the height from which it felt. what is the total vertical distance the ball traveled when it its the ground for the seventh time?

The downward and upward distances form two geometric sequences.

The sum of downward distance is:

s(n)=16(1-.5^n)/(1-.5)

The sum of upward distance is:

s(n)=8(1-.5^(n-1))/(1-.5)

If we let these sums be d(n) and u(n) for upward and downward distances, and neaten them up a bit we have:

d(n)=32(1-.5^n)  and u(n)=16(1-.5^(n-1))

So the total vertical distance traveled when the ball hits the ground for the seventh time is:

D=32(1-.5^7)+16(1-.5^6)

D=32(0.9921875)+16(0.984375)

D=31.75+15.75

D=47.5 ft