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
