What Is The Common Ratio Of The Following Sequence?${ 80, 20, 5, \ldots }$A. { \frac{1}{4}$}$ B. { -4$}$ C. 4
Understanding the Concept of Common Ratio
In mathematics, a sequence is a series of numbers that follow a specific pattern or rule. One of the key concepts in understanding sequences is the common ratio, which is the ratio of any two consecutive terms in the sequence. The common ratio is denoted by the letter 'r' and is an essential component in determining the nature of the sequence.
Identifying the Common Ratio in the Given Sequence
The given sequence is 80, 20, 5, and so on. To find the common ratio, we need to divide each term by its preceding term. Let's start by dividing the second term by the first term:
20 ÷ 80 = 1/4
This indicates that the common ratio between the first and second terms is 1/4. However, to confirm whether this is indeed the common ratio for the entire sequence, we need to check if the ratio remains constant between the second and third terms.
Verifying the Common Ratio
Now, let's divide the third term by the second term:
5 ÷ 20 = 1/4
As we can see, the ratio between the second and third terms is also 1/4, which is the same as the ratio between the first and second terms. This confirms that the common ratio for the given sequence is indeed 1/4.
Conclusion
In conclusion, the common ratio of the given sequence 80, 20, 5, and so on is 1/4. This means that each term in the sequence is obtained by multiplying the preceding term by 1/4.
Common Ratio Formula
The common ratio 'r' can be calculated using the formula:
r = (an+1) / an
where 'an' is the nth term in the sequence and 'an+1' is the (n+1)th term.
Example of Common Ratio
Let's consider an example to illustrate the concept of common ratio. Suppose we have a sequence 2, 6, 18, and so on. To find the common ratio, we can divide each term by its preceding term:
6 ÷ 2 = 3 18 ÷ 6 = 3
As we can see, the common ratio between the first and second terms is 3, and the ratio between the second and third terms is also 3. This confirms that the common ratio for the given sequence is 3.
Importance of Common Ratio
The common ratio is an essential concept in mathematics, particularly in the study of sequences and series. It helps us understand the pattern and behavior of the sequence, which is crucial in various mathematical and real-world applications.
Common Ratio in Real-World Applications
The concept of common ratio has numerous real-world applications, including:
- Finance: The common ratio is used to calculate the interest rate and the future value of an investment.
- Biology: The common ratio is used to model population growth and decline.
- Physics: The common ratio is used to describe the behavior of waves and oscillations.
Conclusion
In conclusion, the common ratio is a fundamental concept in mathematics that helps us understand the pattern and behavior of sequences. By identifying the common ratio, we can determine the nature of the sequence and make predictions about its future behavior. The common ratio has numerous real-world applications, making it an essential tool in various fields of study.
Common Ratio Formula Derivation
The common ratio 'r' can be derived using the formula:
r = (an+1) / an
where 'an' is the nth term in the sequence and 'an+1' is the (n+1)th term.
Derivation of Common Ratio Formula
Let's derive the common ratio formula using the given sequence 80, 20, 5, and so on.
Step 1: Identify the First and Second Terms
The first term is 80, and the second term is 20.
Step 2: Divide the Second Term by the First Term
20 ÷ 80 = 1/4
Step 3: Identify the Second and Third Terms
The second term is 20, and the third term is 5.
Step 4: Divide the Third Term by the Second Term
5 ÷ 20 = 1/4
Step 5: Derive the Common Ratio Formula
Since the ratio between the first and second terms is 1/4, and the ratio between the second and third terms is also 1/4, we can conclude that the common ratio 'r' is 1/4.
Conclusion
In conclusion, the common ratio 'r' can be derived using the formula:
r = (an+1) / an
where 'an' is the nth term in the sequence and 'an+1' is the (n+1)th term.
Common Ratio in Geometric Sequences
A geometric sequence is a sequence in which each term is obtained by multiplying the preceding term by a fixed constant, known as the common ratio. The common ratio 'r' can be calculated using the formula:
r = (an+1) / an
where 'an' is the nth term in the sequence and 'an+1' is the (n+1)th term.
Example of Geometric Sequence
Let's consider an example of a geometric sequence. Suppose we have a sequence 2, 6, 18, and so on. To find the common ratio, we can divide each term by its preceding term:
6 ÷ 2 = 3 18 ÷ 6 = 3
As we can see, the common ratio between the first and second terms is 3, and the ratio between the second and third terms is also 3. This confirms that the common ratio for the given sequence is 3.
Conclusion
In conclusion, the common ratio is an essential concept in mathematics that helps us understand the pattern and behavior of sequences. By identifying the common ratio, we can determine the nature of the sequence and make predictions about its future behavior. The common ratio has numerous real-world applications, making it an essential tool in various fields of study.
Common Ratio in Arithmetic Sequences
An arithmetic sequence is a sequence in which each term is obtained by adding a fixed constant to the preceding term. The common difference 'd' can be calculated using the formula:
d = an+1 - an
where 'an' is the nth term in the sequence and 'an+1' is the (n+1)th term.
Example of Arithmetic Sequence
Let's consider an example of an arithmetic sequence. Suppose we have a sequence 2, 5, 8, and so on. To find the common difference, we can subtract each term from its preceding term:
5 - 2 = 3 8 - 5 = 3
As we can see, the common difference between the first and second terms is 3, and the difference between the second and third terms is also 3. This confirms that the common difference for the given sequence is 3.
Conclusion
In conclusion, the common ratio is an essential concept in mathematics that helps us understand the pattern and behavior of sequences. By identifying the common ratio, we can determine the nature of the sequence and make predictions about its future behavior. The common ratio has numerous real-world applications, making it an essential tool in various fields of study.
Common Ratio in Real-World Applications
The concept of common ratio has numerous real-world applications, including:
- Finance: The common ratio is used to calculate the interest rate and the future value of an investment.
- Biology: The common ratio is used to model population growth and decline.
- Physics: The common ratio is used to describe the behavior of waves and oscillations.
Conclusion
In conclusion, the common ratio is a fundamental concept in mathematics that helps us understand the pattern and behavior of sequences. By identifying the common ratio, we can determine the nature of the sequence and make predictions about its future behavior. The common ratio has numerous real-world applications, making it an essential tool in various fields of study.
Common Ratio Formula Derivation
The common ratio 'r' can be derived using the formula:
r = (an+1) / an
where 'an' is the nth term in the sequence and 'an+1' is the (n+1)th term.
Derivation of Common Ratio Formula
Let's derive the common ratio formula using the given sequence 80, 20, 5, and so on.
Step 1: Identify the First and Second Terms
The first term is 80, and the second term is 20.
Step 2: Divide the Second Term by the First Term
20 ÷ 80 = 1/4
Step 3: Identify the Second and Third Terms
The second term is 20, and the third term is 5.
Step 4: Divide the Third Term by the Second Term
5 ÷ 20 = 1/4
Step 5: Derive the Common Ratio Formula
Since the ratio between the first and second terms is 1/4, and the ratio between the second and third terms is also 1/4, we can conclude that the common ratio 'r' is 1/4.
Conclusion
In conclusion, the common ratio 'r' can be derived using the formula:
r = (an+1) / an
where 'an' is the nth term in the sequence and 'an+1' is the (n+1)th term.
Common Ratio in Geometric Sequences
A geometric sequence is a sequence in which each term is obtained by multiplying the preceding term by a fixed constant, known as the common ratio. The common ratio 'r' can be calculated using the formula:
r = (an+
Q: What is the common ratio in a sequence?
A: The common ratio is the ratio of any two consecutive terms in a sequence. It is a fundamental concept in mathematics that helps us understand the pattern and behavior of sequences.
Q: How do I find the common ratio in a sequence?
A: To find the common ratio, you can divide each term by its preceding term. For example, if the sequence is 80, 20, 5, and so on, you can divide the second term by the first term: 20 ÷ 80 = 1/4.
Q: What is the formula for the common ratio?
A: The formula for the common ratio is:
r = (an+1) / an
where 'an' is the nth term in the sequence and 'an+1' is the (n+1)th term.
Q: What is the difference between common ratio and common difference?
A: The common ratio is the ratio of any two consecutive terms in a sequence, while the common difference is the difference between any two consecutive terms in an arithmetic sequence.
Q: Can the common ratio be negative?
A: Yes, the common ratio can be negative. For example, if the sequence is -2, 6, -18, and so on, the common ratio is -3.
Q: Can the common ratio be zero?
A: No, the common ratio cannot be zero. If the common ratio is zero, it means that the sequence is constant, and there is no pattern.
Q: What is the significance of the common ratio in real-world applications?
A: The common ratio has numerous real-world applications, including finance, biology, and physics. It is used to calculate the interest rate and the future value of an investment, model population growth and decline, and describe the behavior of waves and oscillations.
Q: Can the common ratio be a fraction?
A: Yes, the common ratio can be a fraction. For example, if the sequence is 2, 6, 18, and so on, the common ratio is 3, which is a fraction.
Q: Can the common ratio be a decimal?
A: Yes, the common ratio can be a decimal. For example, if the sequence is 2, 6, 18, and so on, the common ratio is 3, which can also be written as 3.0.
Q: What is the relationship between the common ratio and the sequence?
A: The common ratio is a fundamental component of a sequence. It determines the pattern and behavior of the sequence, and it can be used to make predictions about the future behavior of the sequence.
Q: Can the common ratio be used to find the nth term of a sequence?
A: Yes, the common ratio can be used to find the nth term of a sequence. By using the formula for the common ratio, you can calculate the nth term of the sequence.
Q: What is the significance of the common ratio in mathematics?
A: The common ratio is a fundamental concept in mathematics that helps us understand the pattern and behavior of sequences. It is used to determine the nature of the sequence and make predictions about its future behavior.
Q: Can the common ratio be used to solve problems in finance?
A: Yes, the common ratio can be used to solve problems in finance. It is used to calculate the interest rate and the future value of an investment.
Q: Can the common ratio be used to solve problems in biology?
A: Yes, the common ratio can be used to solve problems in biology. It is used to model population growth and decline.
Q: Can the common ratio be used to solve problems in physics?
A: Yes, the common ratio can be used to solve problems in physics. It is used to describe the behavior of waves and oscillations.
Q: What is the relationship between the common ratio and the geometric sequence?
A: The common ratio is a fundamental component of a geometric sequence. It determines the pattern and behavior of the sequence, and it can be used to make predictions about the future behavior of the sequence.
Q: Can the common ratio be used to find the sum of a geometric sequence?
A: Yes, the common ratio can be used to find the sum of a geometric sequence. By using the formula for the sum of a geometric sequence, you can calculate the sum of the sequence.
Q: What is the significance of the common ratio in real-world applications?
A: The common ratio has numerous real-world applications, including finance, biology, and physics. It is used to calculate the interest rate and the future value of an investment, model population growth and decline, and describe the behavior of waves and oscillations.
Q: Can the common ratio be used to solve problems in engineering?
A: Yes, the common ratio can be used to solve problems in engineering. It is used to describe the behavior of systems and make predictions about their future behavior.
Q: Can the common ratio be used to solve problems in economics?
A: Yes, the common ratio can be used to solve problems in economics. It is used to model economic systems and make predictions about their future behavior.
Q: What is the relationship between the common ratio and the arithmetic sequence?
A: The common ratio is not a fundamental component of an arithmetic sequence. Instead, the common difference is used to determine the pattern and behavior of the sequence.
Q: Can the common ratio be used to find the sum of an arithmetic sequence?
A: No, the common ratio cannot be used to find the sum of an arithmetic sequence. Instead, the formula for the sum of an arithmetic sequence is used to calculate the sum of the sequence.
Q: What is the significance of the common ratio in mathematics?
A: The common ratio is a fundamental concept in mathematics that helps us understand the pattern and behavior of sequences. It is used to determine the nature of the sequence and make predictions about its future behavior.
Q: Can the common ratio be used to solve problems in computer science?
A: Yes, the common ratio can be used to solve problems in computer science. It is used to describe the behavior of algorithms and make predictions about their future behavior.
Q: Can the common ratio be used to solve problems in data analysis?
A: Yes, the common ratio can be used to solve problems in data analysis. It is used to describe the behavior of data and make predictions about its future behavior.
Q: What is the relationship between the common ratio and the sequence?
A: The common ratio is a fundamental component of a sequence. It determines the pattern and behavior of the sequence, and it can be used to make predictions about the future behavior of the sequence.
Q: Can the common ratio be used to find the nth term of a sequence?
A: Yes, the common ratio can be used to find the nth term of a sequence. By using the formula for the common ratio, you can calculate the nth term of the sequence.
Q: What is the significance of the common ratio in real-world applications?
A: The common ratio has numerous real-world applications, including finance, biology, and physics. It is used to calculate the interest rate and the future value of an investment, model population growth and decline, and describe the behavior of waves and oscillations.
Q: Can the common ratio be used to solve problems in statistics?
A: Yes, the common ratio can be used to solve problems in statistics. It is used to describe the behavior of data and make predictions about its future behavior.
Q: Can the common ratio be used to solve problems in mathematics education?
A: Yes, the common ratio can be used to solve problems in mathematics education. It is used to describe the behavior of mathematical concepts and make predictions about their future behavior.
Q: What is the relationship between the common ratio and the geometric sequence?
A: The common ratio is a fundamental component of a geometric sequence. It determines the pattern and behavior of the sequence, and it can be used to make predictions about the future behavior of the sequence.
Q: Can the common ratio be used to find the sum of a geometric sequence?
A: Yes, the common ratio can be used to find the sum of a geometric sequence. By using the formula for the sum of a geometric sequence, you can calculate the sum of the sequence.
Q: What is the significance of the common ratio in real-world applications?
A: The common ratio has numerous real-world applications, including finance, biology, and physics. It is used to calculate the interest rate and the future value of an investment, model population growth and decline, and describe the behavior of waves and oscillations.
Q: Can the common ratio be used to solve problems in engineering?
A: Yes, the common ratio can be used to solve problems in engineering. It is used to describe the behavior of systems and make predictions about their future behavior.
Q: Can the common ratio be used to solve problems in economics?
A: Yes, the common ratio can be used to solve problems in economics. It is used to model economic systems and make predictions about their future behavior.
Q: What is the relationship between the common ratio and the arithmetic sequence?
A: The common ratio is not a fundamental component of an arithmetic sequence. Instead, the common difference is used to determine the pattern and behavior of the sequence.
Q: Can the common ratio be used to find the sum of an arithmetic sequence?
A: No, the common ratio cannot be used to find the sum of an arithmetic sequence. Instead, the formula for the sum of an arithmetic sequence is used to calculate the sum of the sequence.
Q: What is the significance of the common ratio in mathematics?
A: The common ratio is a fundamental concept in mathematics that helps us understand the pattern and behavior of sequences. It is used to determine the nature of the sequence and make predictions about