A Photographer Charges $$ 100 100 100 $ For The First Hour Of A Photoshoot, And The Hourly Rate Is Halved For Each Additional Hour. Which Of The Following Geometric Series Models Represents The Income Of The Photographer After 4 Hours? A.
Introduction
When it comes to calculating the income of a photographer after a certain number of hours, a geometric series can be a useful tool. In this scenario, the photographer charges $100 for the first hour and then halves the hourly rate for each additional hour. We need to determine which geometric series model represents the income of the photographer after 4 hours.
Understanding the Problem
Let's break down the problem step by step. The photographer charges $100 for the first hour, which is the first term of the series. For each additional hour, the hourly rate is halved. This means that the second hour will be $50, the third hour will be $25, and the fourth hour will be $12.50.
Geometric Series Model
A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this case, the common ratio is 1/2, since the hourly rate is halved for each additional hour.
Calculating the Income
To calculate the income of the photographer after 4 hours, we need to sum up the terms of the geometric series. The formula for the sum of a geometric series is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Possible Models
There are several possible geometric series models that could represent the income of the photographer after 4 hours. Let's examine each of them:
Model A
The first model is:
100 + 50 + 25 + 12.5 = 187.5
This model represents the income of the photographer after 4 hours, with the first term being $100, the second term being $50, the third term being $25, and the fourth term being $12.50.
Model B
The second model is:
100 + 50 + 25 + 12.5 + 6.25 = 193.75
This model represents the income of the photographer after 5 hours, with the first term being $100, the second term being $50, the third term being $25, the fourth term being $12.50, and the fifth term being $6.25.
Model C
The third model is:
100 + 50 + 25 + 12.5 + 6.25 + 3.125 = 196.875
This model represents the income of the photographer after 6 hours, with the first term being $100, the second term being $50, the third term being $25, the fourth term being $12.50, the fifth term being $6.25, and the sixth term being $3.125.
Conclusion
Based on the calculations above, the correct geometric series model that represents the income of the photographer after 4 hours is Model A:
100 + 50 + 25 + 12.5 = 187.5
This model accurately represents the income of the photographer after 4 hours, with the first term being $100, the second term being $50, the third term being $25, and the fourth term being $12.50.
Discussion
The geometric series model is a useful tool for calculating the income of a photographer after a certain number of hours. By understanding the problem and the common ratio, we can determine which model represents the income of the photographer after 4 hours. In this case, Model A is the correct model.
Geometric Series Formula
The formula for the sum of a geometric series is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Common Ratio
The common ratio is the fixed, non-zero number that is multiplied by each term to get the next term. In this case, the common ratio is 1/2, since the hourly rate is halved for each additional hour.
Geometric Series Example
A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, the sequence 2, 4, 8, 16, ... is a geometric series with a common ratio of 2.
Geometric Series Applications
Geometric series have many applications in mathematics and other fields. They can be used to model population growth, financial investments, and other real-world phenomena.
Geometric Series Formula Derivation
The formula for the sum of a geometric series can be derived by using the formula for the sum of an infinite geometric series:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Proof
The formula for the sum of a geometric series can be proved by using the formula for the sum of an infinite geometric series:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Extension
The formula for the sum of a geometric series can be extended to include the case where the series has a finite number of terms. In this case, the formula is:
S = a * (1 - r^n) / (1 - r)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
Geometric Series Formula Simplification
The formula for the sum of a geometric series can be simplified by using the formula for the sum of an infinite geometric series:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Generalization
The formula for the sum of a geometric series can be generalized to include the case where the series has a non-constant common ratio. In this case, the formula is:
S = a * (1 - r^n) / (1 - r)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
Geometric Series Formula Special Case
The formula for the sum of a geometric series has a special case where the common ratio is 1. In this case, the formula is:
S = a * n
where S is the sum, a is the first term, and n is the number of terms.
Geometric Series Formula Limit
The formula for the sum of a geometric series has a limit as the number of terms approaches infinity. In this case, the formula is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Inequality
The formula for the sum of a geometric series has an inequality that bounds the sum. In this case, the inequality is:
S ≤ a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Convergence
The formula for the sum of a geometric series converges to a finite value as the number of terms approaches infinity. In this case, the formula is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Divergence
The formula for the sum of a geometric series diverges to infinity as the number of terms approaches infinity. In this case, the formula is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Stability
The formula for the sum of a geometric series is stable with respect to small changes in the common ratio. In this case, the formula is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Sensitivity
The formula for the sum of a geometric series is sensitive to large changes in the common ratio. In this case, the formula is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Robustness
The formula for the sum of a geometric series is robust with respect to outliers in the data. In this case, the formula is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Accuracy
The formula for the sum of a geometric series is accurate for a wide range of values of the common ratio. In this case, the formula is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Precision
The formula for the sum of a geometric series is precise for a wide range of values of the common ratio. In this case, the formula is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Reliability
The formula for the sum of a geometric series is reliable for a wide range of values of the common ratio. In this case, the formula is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Geometric Series Formula Validity
The formula for the sum of a geometric series is valid for a wide range of values of the common ratio. In this case, the formula is:
S = a / (1 - r)
where S is the sum
Introduction
In our previous article, we discussed how a geometric series can be used to model the income of a photographer after a certain number of hours. We also examined several possible geometric series models that could represent the income of the photographer after 4 hours. In this article, we will answer some frequently asked questions about geometric series and their applications in modeling the income of a photographer.
Q: What is a geometric series?
A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Q: What is the common ratio in a geometric series?
A: The common ratio is the fixed, non-zero number that is multiplied by each term to get the next term. In the case of the photographer's income, the common ratio is 1/2, since the hourly rate is halved for each additional hour.
Q: How do I calculate the sum of a geometric series?
A: The formula for the sum of a geometric series is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Q: What is the significance of the common ratio in a geometric series?
A: The common ratio determines the rate at which the terms of the series increase or decrease. In the case of the photographer's income, the common ratio of 1/2 means that the hourly rate is halved for each additional hour.
Q: Can a geometric series have a negative common ratio?
A: Yes, a geometric series can have a negative common ratio. However, in the case of the photographer's income, the common ratio is positive, since the hourly rate is not decreasing.
Q: Can a geometric series have a common ratio of 1?
A: Yes, a geometric series can have a common ratio of 1. However, in the case of the photographer's income, the common ratio is not 1, since the hourly rate is decreasing.
Q: What is the difference between a geometric series and an arithmetic series?
A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. An arithmetic series, on the other hand, is a sequence of numbers in which each term after the first is found by adding a fixed number called the common difference.
Q: Can a geometric series be used to model the income of a photographer after a certain number of hours?
A: Yes, a geometric series can be used to model the income of a photographer after a certain number of hours. In this case, the common ratio is 1/2, since the hourly rate is halved for each additional hour.
Q: What are some real-world applications of geometric series?
A: Geometric series have many real-world applications, including modeling population growth, financial investments, and other real-world phenomena.
Q: Can a geometric series be used to model the income of a photographer after a certain number of hours with a non-constant common ratio?
A: Yes, a geometric series can be used to model the income of a photographer after a certain number of hours with a non-constant common ratio. However, in this case, the formula for the sum of a geometric series would need to be modified to account for the non-constant common ratio.
Q: What is the significance of the first term in a geometric series?
A: The first term in a geometric series determines the starting value of the series. In the case of the photographer's income, the first term is $100, which is the hourly rate for the first hour.
Q: Can a geometric series have a first term of 0?
A: Yes, a geometric series can have a first term of 0. However, in the case of the photographer's income, the first term is not 0, since the hourly rate is not 0.
Q: What is the difference between a geometric series and a harmonic series?
A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. A harmonic series, on the other hand, is a sequence of numbers in which each term after the first is found by taking the reciprocal of the previous term.
Q: Can a geometric series be used to model the income of a photographer after a certain number of hours with a harmonic common ratio?
A: No, a geometric series cannot be used to model the income of a photographer after a certain number of hours with a harmonic common ratio. However, a harmonic series could be used to model the income of a photographer after a certain number of hours with a harmonic common ratio.
Q: What is the significance of the number of terms in a geometric series?
A: The number of terms in a geometric series determines the length of the series. In the case of the photographer's income, the number of terms is 4, since we are modeling the income after 4 hours.
Q: Can a geometric series have a finite number of terms?
A: Yes, a geometric series can have a finite number of terms. However, in the case of the photographer's income, the number of terms is not finite, since we are modeling the income after an infinite number of hours.
Q: What is the difference between a geometric series and a power series?
A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. A power series, on the other hand, is a sequence of numbers in which each term after the first is found by raising the previous term to a fixed power.
Q: Can a geometric series be used to model the income of a photographer after a certain number of hours with a power common ratio?
A: No, a geometric series cannot be used to model the income of a photographer after a certain number of hours with a power common ratio. However, a power series could be used to model the income of a photographer after a certain number of hours with a power common ratio.
Q: What is the significance of the common ratio in a geometric series?
A: The common ratio determines the rate at which the terms of the series increase or decrease. In the case of the photographer's income, the common ratio of 1/2 means that the hourly rate is halved for each additional hour.
Q: Can a geometric series have a common ratio of 0?
A: No, a geometric series cannot have a common ratio of 0. However, a geometric series can have a common ratio that approaches 0 as the number of terms approaches infinity.
Q: What is the difference between a geometric series and a binomial series?
A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. A binomial series, on the other hand, is a sequence of numbers in which each term after the first is found by raising the previous term to a fixed power.
Q: Can a geometric series be used to model the income of a photographer after a certain number of hours with a binomial common ratio?
A: No, a geometric series cannot be used to model the income of a photographer after a certain number of hours with a binomial common ratio. However, a binomial series could be used to model the income of a photographer after a certain number of hours with a binomial common ratio.
Q: What is the significance of the first term in a geometric series?
A: The first term in a geometric series determines the starting value of the series. In the case of the photographer's income, the first term is $100, which is the hourly rate for the first hour.
Q: Can a geometric series have a first term of 1?
A: Yes, a geometric series can have a first term of 1. However, in the case of the photographer's income, the first term is not 1, since the hourly rate is not 1.
Q: What is the difference between a geometric series and a Fibonacci series?
A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. A Fibonacci series, on the other hand, is a sequence of numbers in which each term after the first is found by adding the previous two terms.
Q: Can a geometric series be used to model the income of a photographer after a certain number of hours with a Fibonacci common ratio?
A: No, a geometric series cannot be used to model the income of a photographer after a certain number of hours with a Fibonacci common ratio. However, a Fibonacci series could be used to model the income of a photographer after a certain number of hours with a Fibonacci common ratio.
Q: What is the significance of the number of terms in a geometric series?
A: The number of terms in a geometric series determines the length of the series. In the case of the photographer's income, the number of terms is 4, since we are modeling the income after 4 hours.
Q: Can a geometric series have a finite number of terms?
A: Yes, a geometric series can have a finite number of terms. However, in the case of the photographer's income, the number of terms is not finite, since we are modeling the income after an infinite number of hours.
Q: What is the difference between a geometric series and a Pascal's triangle series?
A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the