A ball is dropped from a height of 16 feet. each time it is drops, it rebounds 80% of the height from which it is falling fine the total distance traveled in 15 bounces

16+2*16 * (0.8) * [ (1 - (0.8) ^15) / (1-0.8) ]=139.5ft

Notice total distance is comprised of both positive movements and negative movements, and both sequences are geometric (exponential) sequences ...

The sum of a geometric sequence is:

s (n) = a (1-r^n) / (1-r), a=initial term, r=common ratio, n=term number

So for the "dropping" distances you have the sum ...

s (n) = 16 (1-.8^n) / 0.2=80 (1-.8^n)

s (15) = 80 (1-.8^15)

And the "rising" distances you the first term is. 8 (16) = 12.8 and n=14 so

s (14) = 12.8 (1-.8^14) / (.2)

s (14) = 64 (1-.8^14)

So the total distance traveled is:

80 (1-.8^15) + 64 (1-.8^14)

138.37050046578688

Total distance is approximately 138.37 ft