SOLUTION: a ball is dropped from a height of 2 meters and bounces up to 3/4 of its previous height on each bounce. using geometric series, how much TOTAL DISTANCE (up and down) does the ball

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Question 1158664: a ball is dropped from a height of 2 meters and bounces up to 3/4 of its previous height on each bounce. using geometric series, how much TOTAL DISTANCE (up and down) does the ball travel until it stops bouncing?
Found 2 solutions by Shin123, ikleyn:
Answer by Shin123(626)   (Show Source): You can put this solution on YOUR website!
The ball keeps bouncing a fraction of its height. The ball in theory never stops bouncing (but in real life, it eventually does). So it is an infinite geometric sequence.
Solved by pluggable solver: Finding Sum of An Infinite Geometric Sequence

So for an infinite geometric series, the sum of it is always under the condition that . In this case, a is 2 and r is 0.75.So the sum of the series is .

Answer by ikleyn(52780)   (Show Source): You can put this solution on YOUR website!
.

In this problem, the sequence is close to be a Geometric progression, but actually, is not.


It requires an ACCURATE analysis. First time the ball falls down; then every next bouncing, it goes UP and DOWN


The total distance is the sum of these values

            Down   Up and       Up and       Up and       Up and 
                    down         down         down         down

              2                           

Coefficient   1       2            2            2           2    <<<---=== this coefficient accounts for "up and down"


So the total distance is

   2   +  2*S,   where S is the infinite geometric progression with the first term a =  =   and the common ratio r = .


The sum of this infinite progression is  S =  =  =  = 6.


Therefore, the ANSWER  to the problem question is 2 + 2*6 = 14 meters.

Solved.

-------------

So, this problem has a "trap".

Those who have experience in solving/teaching/explaining such problems, know about it.

All the others, as a rule, go directly to this trap.


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