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

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

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