Applying Geometric Sequences to the Problem

Oct 13, 2025 by leolavanderia.com.br 10 views

Understanding the Problem: A Ball's Descent and the Concept of Geometric Sequences

When a ball is dropped from a certain height, it follows a curved path under the influence of gravity. The height of the ball at each point in time can be described using a geometric sequence. In this article, we will explore the concept of geometric sequences and apply it to the problem of a ball being dropped from a height of 0.5 meters.

What is a Geometric Sequence?

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general formula for a geometric sequence is:

a_n = a_1 * r^(n-1)

where a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the term number.

Applying Geometric Sequences to the Problem

In this problem, we are given that each curved path reaches 52% of the height of the previous path. This means that the common ratio (r) is 0.52. We are also given that the initial height is 0.5 meters, which corresponds to the first term (a_1) of the sequence.

Writing a Rule for the Sequence

To write a rule for the sequence, we need to express the height of the ball at each point in time using the formula for a geometric sequence. Since the initial height is given by the term n=1, we can start by substituting n=1 into the formula:

a_1 = a_1 * r^(1-1) a_1 = a_1 * r^0 a_1 = a_1

This confirms that the initial height is indeed a_1.

Now, we can substitute n=2 into the formula to find the height of the ball at the second point in time:

a_2 = a_1 * r^(2-1) a_2 = a_1 * r^1 a_2 = a_1 * 0.52

We can continue this process to find the height of the ball at each subsequent point in time.

Finding the Height of the Ball at Each Point in Time

Using the formula for a geometric sequence, we can find the height of the ball at each point in time:

n	a_n
1	0.5
2	0.5 * 0.52
3	0.5 * (0.52)^2
4	0.5 * (0.52)^3
5	0.5 * (0.52)^4

We can see that the height of the ball at each point in time is given by the formula:

a_n = 0.5 * (0.52)^(n-1)

Converting the Height to Centimeters

Since the problem asks us to write a rule for the sequence using centimeters, we need to convert the height from meters to centimeters. There are 100 centimeters in 1 meter, so we can multiply the height in meters by 100 to get the height in centimeters:

a_n = 0.5 * (0.52)^(n-1) * 100

Simplifying the Formula

We can simplify the formula by multiplying the constants:

a_n = 50 * (0.52)^(n-1)

What Height Will the Ball Be at Each Point in Time?

Using the formula, we can find the height of the ball at each point in time:

n	a_n
1	50
2	50 * 0.52
3	50 * (0.52)^2
4	50 * (0.52)^3
5	50 * (0.52)^4

We can see that the height of the ball at each point in time is given by the formula:

a_n = 50 * (0.52)^(n-1)

Conclusion

In this article, we explored the concept of geometric sequences and applied it to the problem of a ball being dropped from a height of 0.5 meters. We found that the height of the ball at each point in time is given by the formula:

a_n = 50 * (0.52)^(n-1)

This formula can be used to find the height of the ball at each point in time, and it can be applied to any problem involving a geometric sequence.
Q&A: Understanding Geometric Sequences and the Ball's Descent

In our previous article, we explored the concept of geometric sequences and applied it to the problem of a ball being dropped from a height of 0.5 meters. In this article, we will answer some frequently asked questions about geometric sequences and the ball's descent.

Q: What is a geometric sequence?

A: A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: How do I find the common ratio in a geometric sequence?

A: To find the common ratio, you need to divide each term by the previous term. For example, if the sequence is 2, 6, 18, 54, ..., the common ratio is 3, since each term is 3 times the previous term.

Q: How do I write a rule for a geometric sequence?

A: To write a rule for a geometric sequence, you need to use the formula:

a_n = a_1 * r^(n-1)

where a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the term number.

Q: What is the initial height of the ball in the problem?

A: The initial height of the ball is 0.5 meters, which corresponds to the first term (a_1) of the sequence.

Q: What is the common ratio in the problem?

A: The common ratio in the problem is 0.52, since each curved path reaches 52% of the height of the previous path.

Q: How do I find the height of the ball at each point in time?

A: To find the height of the ball at each point in time, you can use the formula:

a_n = 50 * (0.52)^(n-1)

where n is the term number.

Q: What is the height of the ball at each point in time?

A: The height of the ball at each point in time is given by the formula:

n	a_n
1	50
2	50 * 0.52
3	50 * (0.52)^2
4	50 * (0.52)^3
5	50 * (0.52)^4

Q: Can I use the formula to find the height of the ball at any point in time?

A: Yes, you can use the formula to find the height of the ball at any point in time. Simply substitute the value of n into the formula and calculate the result.

Q: What is the significance of the common ratio in the problem?

A: The common ratio in the problem represents the percentage of the height of the previous path that the ball reaches at each point in time. In this case, the common ratio is 0.52, which means that the ball reaches 52% of the height of the previous path at each point in time.

Q: Can I apply the concept of geometric sequences to other problems?

A: Yes, you can apply the concept of geometric sequences to other problems involving exponential growth or decay. For example, you can use geometric sequences to model population growth, financial investments, or chemical reactions.

Conclusion

In this article, we answered some frequently asked questions about geometric sequences and the ball's descent. We hope that this article has helped you to understand the concept of geometric sequences and how to apply it to real-world problems.