Here’s a word problem involving the sum of an infinite geometric sequence:

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Word Problem:

A ball is dropped from a height of 10 meters. Each time it hits the ground, it bounces back to a height that is 60% of the previous height. The ball continues to bounce indefinitely, with each bounce reaching a smaller height.

What is the total distance the ball travels by the time it stops bouncing?

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Hint: The distance the ball travels includes both the falling and bouncing back parts. The first drop is 10 meters, and each subsequent bounce up and down forms part of an infinite geometric sequence.

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Would you like a detailed explanation of how to solve this?

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Here are 5 related questions to expand on this:

1. What if the ball bounces back to 80% of its height instead of 60%?
2. How would the total distance change if the initial height was 20 meters?
3. What if the ball stops bouncing after 10 bounces?
4. Can you model this problem with an arithmetic sequence instead? Why or why not?
5. How does the rate of decay (percentage of bounce height) affect the sum?

Tip: In geometric sequences, if the common ratio is between -1 and 1, the infinite sum formula is $S = \frac{a}{1 - r}$.