SOLUTION: I've tried different ways to write an equation but am having no luck. Can anyone help? A tennis ball dropped onto a hard surface rebounds exactly 2/3 the height from which it f

Question 177428: I've tried different ways to write an equation but am having no luck. Can anyone help?
A tennis ball dropped onto a hard surface rebounds exactly 2/3 the height from which it falls. How far will it rebound after hitting the surface for the third time if it is dropped from an initial height of 18 feet?
Found 2 solutions by jim_thompson5910, solver91311:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
After the first drop, it rebounds back to the height of feet. So it starts at 18 ft, drops to 0 ft, then bounces back to 12 ft

Now after the second drop, it bounces back to a height of ft. So the ball starts at 12 ft, goes to 0 ft, then back up to 8 ft

Finally, after the third drop, the ball rebounds to a height of about (note: this is approximate) feet. So it starts at 8 ft, drops to 0 ft, and then back up to roughly 5.333 ft

So the answer is 5.333 ft

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Notice how I took each previous answer and I multiplied it by . So on the first drop, I multiplied by 18 to get the following (Let's call that result "a")

So

Now I then multiplied that result by another to get (now let's call it "b")

Plug in

Finally, I multiplied that previous result "b" by to get the next height (which we'll call "c")

Plug in

So after the third drop, there are 3 terms. After the second drop, there are 2 terms. After the first drop, there's only one term.

So this means that after "x" drops (where you can plug any positive number in for "x"), there will be "x" terms after the initial "18". So the equation might look like this:

To make things look tidy, we can condense the "x" number of terms to get

===============================================

Answer:

So the equation is where "h" is the height and "x" is the number of drops.

Notice how if we let , then

which was our original answer.

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
The first time it falls, it rebounds to if is the initial height. For the second time, is no longer the initial height, is, so the rebound height becomes or . In general, the rebound height for the th bounce is . Therefore, you need to calculate
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