A Bouncing Toy Reaches A Height Of 64 Inches At Its First Peak, 48 Inches At Its Second Peak, And 36 Inches At Its Third Peak. Which Explicit Function Represents The Geometric Sequence Of The Heights Of The Toy?A.
Introduction
A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this article, we will explore the geometric sequence of the heights of a bouncing toy, which reaches a height of 64 inches at its first peak, 48 inches at its second peak, and 36 inches at its third peak. We will determine the explicit function that represents this geometric sequence.
Understanding Geometric Sequences
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 of the sequence
- r is the common ratio
- n is the term number
Finding the Common Ratio
To find the common ratio, we can divide any term by its previous term. Let's use the first two terms:
r = a_2 / a_1 = 48 / 64 = 0.75
So, the common ratio is 0.75.
Determining the Explicit Function
Now that we have the common ratio, we can determine the explicit function that represents the geometric sequence. The general formula for a geometric sequence is:
a_n = a_1 * r^(n-1)
We know that a_1 = 64 and r = 0.75. Plugging these values into the formula, we get:
a_n = 64 * (0.75)^(n-1)
This is the explicit function that represents the geometric sequence of the heights of the toy.
Verifying the Function
To verify that this function represents the geometric sequence, we can plug in the values of n for the first three terms:
a_1 = 64 * (0.75)^0 = 64 a_2 = 64 * (0.75)^1 = 48 a_3 = 64 * (0.75)^2 = 36
As expected, the function produces the correct values for the first three terms.
Conclusion
In this article, we have determined the explicit function that represents the geometric sequence of the heights of a bouncing toy. The function is:
a_n = 64 * (0.75)^(n-1)
This function can be used to find the height of the toy at any peak, given the peak number. We have also verified that this function produces the correct values for the first three terms.
Example Use Cases
This function can be used in a variety of situations, such as:
- Predicting the height of a bouncing toy at a future peak
- Determining the number of peaks a toy will reach before it comes to rest
- Modeling the behavior of a bouncing toy in a simulation
Code Implementation
Here is an example of how to implement this function in Python:
def bouncing_toy_height(n):
"""
Returns the height of a bouncing toy at the nth peak.
Parameters:
n (int): The peak number.
Returns:
float: The height of the toy at the nth peak.
"""
a_1 = 64
r = 0.75
return a_1 * (r ** (n - 1))
This function takes an integer n as input and returns the height of the toy at the nth peak.
References
Introduction
In our previous article, we explored the geometric sequence of the heights of a bouncing toy, which reaches a height of 64 inches at its first peak, 48 inches at its second peak, and 36 inches at its third peak. We determined the explicit function that represents this geometric sequence. In this article, we will answer some frequently asked questions about the geometric sequence of the heights of a bouncing toy.
Q: What is a geometric sequence?
A: A geometric sequence is a type of sequence 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 of a geometric sequence?
A: To find the common ratio, you can divide any term by its previous term. For example, if the first two terms are 64 and 48, the common ratio would be 48/64 = 0.75.
Q: What is the explicit function that represents the geometric sequence of the heights of a bouncing toy?
A: The explicit function that represents the geometric sequence of the heights of a bouncing toy is:
a_n = 64 * (0.75)^(n-1)
Q: How do I use the explicit function to find the height of the toy at a future peak?
A: To use the explicit function to find the height of the toy at a future peak, simply plug in the peak number (n) into the function. For example, to find the height of the toy at the 5th peak, you would plug in n = 5:
a_5 = 64 * (0.75)^(5-1) = 64 * (0.75)^4 = 16.39
Q: Can I use the explicit function to model the behavior of a bouncing toy in a simulation?
A: Yes, you can use the explicit function to model the behavior of a bouncing toy in a simulation. The function can be used to predict the height of the toy at any peak, given the peak number.
Q: What are some real-world applications of geometric sequences?
A: Geometric sequences have many real-world applications, including:
- Modeling population growth
- Predicting stock prices
- Analyzing the behavior of bouncing objects
- Determining the number of peaks a toy will reach before it comes to rest
Q: Can I use the explicit function to find the number of peaks a toy will reach before it comes to rest?
A: Yes, you can use the explicit function to find the number of peaks a toy will reach before it comes to rest. To do this, you would need to find the value of n for which the height of the toy is approximately 0. This can be done using a calculator or computer program.
Q: What is the difference between an explicit function and an implicit function?
A: An explicit function is a function where the output (y) is expressed in terms of the input (x). An implicit function is a function where the output (y) is not expressed in terms of the input (x), but rather in terms of a relationship between the input and output.
Q: Can I use the explicit function to find the height of a toy that is not bouncing?
A: No, the explicit function is only applicable to bouncing toys. If you want to find the height of a toy that is not bouncing, you would need to use a different function.
Q: What are some common mistakes to avoid when working with geometric sequences?
A: Some common mistakes to avoid when working with geometric sequences include:
- Not checking for the existence of a common ratio
- Not checking for the existence of a finite number of terms
- Not using the correct formula for the nth term
- Not checking for the existence of a limit
Conclusion
In this article, we have answered some frequently asked questions about the geometric sequence of the heights of a bouncing toy. We have also discussed some real-world applications of geometric sequences and some common mistakes to avoid when working with them.