I need some help with these quesitons
1. a1 = -1, and an = 4/3an-1 for n = 2, 3, 4, ___. What is the common ration, r, for this sequence?
2. A rubber ball is dropped from a height of 50 ft. Assume that each bounce of the ball is vertical and that each bounce reaches a height of 5/6 of the maximum height of the previous bounce. What is the total distance traveled throughout the lifetime of bounces for this ball?
I'm not sure I even understand this question
3. The following infinite geometric series will have a finite sum: 1001/6, 1001/36, 1001/216, 1001/1296
true?
Sure, I can help you with these questions. Let's go through each question one by one and explain how to find the answers.
1. For the given sequence, a1 = -1 and an = (4/3) * an-1 for n = 2, 3, 4, _____. To find the common ratio, we need to determine the ratio between any two consecutive terms. In this case, let's consider a2 and a1.
To find a2, we substitute n = 2 into the formula:
a2 = (4/3) * a1
= (4/3) * (-1)
= -4/3
Now, we can determine the common ratio, r, by dividing a2 by a1:
r = a2 / a1
= (-4/3) / (-1)
= 4/3
Therefore, the common ratio, r, for this sequence is 4/3.
2. The question involves a rubber ball being dropped from a height of 50 ft and each bounce reaching 5/6 of the maximum height of the previous bounce. To find the total distance traveled throughout the lifetime of bounces, we need to consider how far the ball travels for each bounce.
When the ball is dropped, it falls from a height of 50 ft. This is considered as the first bounce. Let's assume this distance as d1.
To find d1, we can calculate the distance traveled during the first fall, which is 50 ft.
Now, for subsequent bounces, each bounce reaches a height of 5/6 times the maximum height of the previous bounce. Let's denote this as h_max.
To find h_max, we multiply 5/6 by the maximum height of the previous bounce. In this case, since the ball was dropped, the maximum height of the previous bounce is 50 ft.
h_max = (5/6) * 50
= 250/6
= 125/3 ft
Now, to find the distance traveled during each bounce, we need to consider the distance from the maximum height to the point of impact. This distance is twice the maximum height, as the ball reaches the maximum height twice during each bounce (once going up, and once coming down).
Therefore, the distance traveled during each bounce is 2 * h_max = 2 * (125/3) ft = 250/3 ft.
Now, to find the total distance traveled throughout the lifetime of bounces, we need to sum up the distances traveled during each bounce, starting from the initial drop (d1).
Since there is an infinite number of bounces, we can sum up this infinite series using the formula for the sum of an infinite geometric series:
Sum = d1 / (1 - r)
where d1 is the initial term and r is the common ratio. In this case, d1 is 50 and we already determined the common ratio as 5/6.
Using the formula, we have:
Sum = 50 / (1 - (5/6))
= 50 / (1/6)
= 300 ft
Therefore, the total distance traveled throughout the lifetime of bounces for this ball is 300 ft.
3. The question is asking if the given infinite geometric series will have a finite sum. To determine this, we need to check the absolute value of the common ratio. If the absolute value of the common ratio is less than 1, the series will have a finite sum.
Let's find the common ratio for this series. We can calculate the ratio between any two consecutive terms. In this case, let's consider the ratio between the second term and the first term.
To find the common ratio, r, we divide the second term by the first term:
r = (1001/36) / (1001/6)
= (1001/36) * (6/1001)
= 6/36
= 1/6
The absolute value of the common ratio, |r|, is 1/6, which is less than 1. Therefore, the given infinite geometric series will have a finite sum.
I hope this explanation helps you understand how to solve these questions. Let me know if you have any further questions!