Understanding the Pattern
* Fall: The ball first falls 8 meters.
* Bounce 1: It bounces back up 8 / 2 = 4 meters and then falls another 4 meters.
* Bounce 2: It bounces back up 4 / 2 = 2 meters and then falls another 2 meters.
* And so on... This pattern continues, with each bounce half the height of the previous one.
Calculating the Total Distance
This forms an infinite geometric series:
8 + (4 + 4) + (2 + 2) + (1 + 1) + ...
To find the total distance, we need to consider:
* The sum of the falling distances: 8 + 4 + 2 + 1 + ...
* The sum of the bouncing distances: 4 + 2 + 1 + ...
Formula for Infinite Geometric Series
The sum (S) of an infinite geometric series is:
S = a / (1 - r)
Where:
* a = the first term
* r = the common ratio (the factor by which each term is multiplied)
Applying the Formula
* Falling Distance:
* a = 8
* r = 1/2
* S = 8 / (1 - 1/2) = 8 / (1/2) = 16 meters
* Bouncing Distance:
* a = 4
* r = 1/2
* S = 4 / (1 - 1/2) = 4 / (1/2) = 8 meters
Total Distance
The total distance traveled is the sum of the falling and bouncing distances:
16 meters + 8 meters = 24 meters
Therefore, the ball will travel a total of 24 meters.
