A ball is dropped vertically. It reaches a height of 1.6m on the first bounce. The height of each subsequent bounce is 80% of the previous bounce.
A. Find the height the ball reaches on the 6th bounce.
B. Find the sum of the first seven terms of this sequence.
A: 1.6 * 0.8^5
B: 1.6 * (1 - 0.8^7)/(1-0.8)
To find the height the ball reaches on the 6th bounce (Question A), we can start by finding the height of the first bounce and then calculate each subsequent bounce using the given information.
1st bounce: 1.6m (given)
2nd bounce: To find the height of the second bounce, we multiply the height of the first bounce (1.6m) by 80% (0.8):
Height of 2nd bounce = 1.6m * 0.8 = 1.28m
3rd bounce: To find the height of the third bounce, we multiply the height of the second bounce (1.28m) by 80% (0.8):
Height of 3rd bounce = 1.28m * 0.8 = 1.024m
4th bounce: Height of 4th bounce = 1.024m * 0.8 = 0.8192m
5th bounce: Height of 5th bounce = 0.8192m * 0.8 = 0.65536m
6th bounce: Height of 6th bounce = 0.65536m * 0.8 = 0.524288m
Therefore, the height the ball reaches on the 6th bounce is approximately 0.524288 meters.
Now, let's move on to Question B, which asks us to find the sum of the first seven terms of this sequence.
To find the sum of the first seven terms, we need to calculate the heights of the first seven bounces and then add them up:
Height of 1st bounce = 1.6m
Height of 2nd bounce = 1.6m * 0.8 = 1.28m
Height of 3rd bounce = 1.28m * 0.8 = 1.024m
Height of 4th bounce = 1.024m * 0.8 = 0.8192m
Height of 5th bounce = 0.8192m * 0.8 = 0.65536m
Height of 6th bounce = 0.65536m * 0.8 = 0.524288m
Height of 7th bounce = 0.524288m * 0.8 = 0.41943m
Now add up all seven heights:
1.6m + 1.28m + 1.024m + 0.8192m + 0.65536m + 0.524288m + 0.41943m = 5.266288m
Therefore, the sum of the first seven terms of this sequence (heights of the first seven bounces) is approximately 5.27 meters.