SOLUTION: A Super Happy Fun Ball is dropped from a height of 8 feet and rebounds 12/13 of the distance from which it fell. How many times will it bounce before its rebound is less than 1

Bounce Number         Rebound
                
                
      .
      .
      .
                 

We set the nth bounce < 1



Multiply both sides by 



Take logs of both sides:





Divide both sides by 

However since 

we are dividing an inequality through by a negative number
and that will reverse the inequality from < to >:








So the 26th rebound will be the first one less than 1 foot.

How far will the ball travel before it comes to rest on the ground? 

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Theoretically it will never stop bouncing, but that's theoretically,
not actually.

However we treat it as an infinite series.

The first 8 feet that it fell must be added in separately.

It rebounded 96/13 feet and then fell 96/13 before it hit the ground
the second time.  So the first term is 2×(96/13) = 192/13 and the common
ratio is 12/13



To that we must add the 8 feet it fell in the beginning:

Answer: 192+8 = 200 feet.

Edwin