What Is The Common Ratio Of The Sequence Below?${ \frac{2}{3}, \frac{1}{6}, \frac{1}{24}, \frac{1}{96}, \ldots }$A. { \frac{1}{9}$}$ B. { \frac{1}{4}$}$ C. 4 D. 9

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What is the Common Ratio of the Sequence Below?

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 concept in algebra and geometry.

The Given Sequence

The given sequence is 23,16,124,196,…\frac{2}{3}, \frac{1}{6}, \frac{1}{24}, \frac{1}{96}, \ldots. To find the common ratio, we need to examine the relationship between consecutive terms in the sequence.

Calculating the Common Ratio

To calculate the common ratio, we can divide any two consecutive terms in the sequence. Let's take the first two terms: 23\frac{2}{3} and 16\frac{1}{6}. We can divide the second term by the first term to get the common ratio:

1623=16×32=312=14\frac{\frac{1}{6}}{\frac{2}{3}} = \frac{1}{6} \times \frac{3}{2} = \frac{3}{12} = \frac{1}{4}

Analyzing the Sequence

Now that we have found the common ratio, let's analyze the sequence to see if it holds true for all consecutive terms. We can take the second and third terms: 16\frac{1}{6} and 124\frac{1}{24}. We can divide the third term by the second term to get the common ratio:

12416=124×61=624=14\frac{\frac{1}{24}}{\frac{1}{6}} = \frac{1}{24} \times \frac{6}{1} = \frac{6}{24} = \frac{1}{4}

Conclusion

Based on our analysis, we can conclude that the common ratio of the given sequence is 14\frac{1}{4}. This means that each term in the sequence is obtained by multiplying the previous term by 14\frac{1}{4}.

Answer

The correct answer is B. 14\frac{1}{4}.

Understanding the Significance of Common Ratio

The common ratio is an essential concept in mathematics, particularly in algebra and geometry. It helps us understand the relationship between consecutive terms in a sequence and is used to find the sum of an infinite geometric series. In this article, we have seen how to calculate the common ratio of a given sequence and how it can be used to analyze the sequence.

Real-World Applications of Common Ratio

The common ratio has numerous real-world applications, including:

  • Finance: The common ratio is used to calculate the interest rate on a loan or investment.
  • Engineering: The common ratio is used to design and analyze electrical circuits and mechanical systems.
  • Computer Science: The common ratio is used in algorithms and data structures to analyze and optimize performance.

Conclusion

In conclusion, the common ratio is an essential concept in mathematics that helps us understand the relationship between consecutive terms in a sequence. By calculating the common ratio, we can analyze the sequence and make predictions about future terms. The common ratio has numerous real-world applications, making it an essential tool in various fields.

Frequently Asked Questions

  • What is the common ratio of a sequence? The common ratio of a sequence is the ratio of any two consecutive terms in the sequence.
  • How do I calculate the common ratio? To calculate the common ratio, divide any two consecutive terms in the sequence.
  • What are the real-world applications of common ratio? The common ratio has numerous real-world applications, including finance, engineering, and computer science.

References

  • Algebra and Geometry: A comprehensive textbook on algebra and geometry that covers the concept of common ratio.
  • Mathematics for Engineers: A textbook on mathematics for engineers that covers the application of common ratio in engineering.
  • Introduction to Computer Science: A textbook on computer science that covers the application of common ratio in algorithms and data structures.
    Frequently Asked Questions: Common Ratio

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 concept in algebra and geometry.

Q&A: Common Ratio

Q: What is the common ratio of a sequence? A: The common ratio of a sequence is the ratio of any two consecutive terms in the sequence.

Q: How do I calculate the common ratio? A: To calculate the common ratio, divide any two consecutive terms in the sequence.

Q: What are the real-world applications of common ratio? A: The common ratio has numerous real-world applications, including finance, engineering, and computer science.

Q: What is the formula for calculating the common ratio? A: The formula for calculating the common ratio is:

r = (a_n+1 / a_n)

where r is the common ratio, a_n is the nth term in the sequence, and a_n+1 is the (n+1)th term in the sequence.

Q: How do I determine if a sequence is geometric? A: A sequence is geometric if the ratio of any two consecutive terms is constant. To determine if a sequence is geometric, calculate the common ratio and check if it is constant.

Q: What is the significance of the common ratio in finance? A: The common ratio is used to calculate the interest rate on a loan or investment. It helps investors and lenders understand the potential returns on their investments.

Q: How do I use the common ratio in engineering? A: The common ratio is used in engineering to design and analyze electrical circuits and mechanical systems. It helps engineers understand the behavior of complex systems and make predictions about their performance.

Q: What is the relationship between the common ratio and the sum of an infinite geometric series? A: The common ratio is used to calculate the sum of an infinite geometric series. The formula for the sum of an infinite geometric series is:

S = a / (1 - r)

where S is the sum of the series, a is the first term, and r is the common ratio.

Q: How do I apply the common ratio in computer science? A: The common ratio is used in computer science to analyze and optimize algorithms and data structures. It helps programmers understand the performance of their code and make predictions about its behavior.

Conclusion

In conclusion, the common ratio is an essential concept in mathematics that has numerous real-world applications. By understanding the concept of common ratio, we can analyze sequences, make predictions about future terms, and apply the concept in various fields.

References

  • Algebra and Geometry: A comprehensive textbook on algebra and geometry that covers the concept of common ratio.
  • Mathematics for Engineers: A textbook on mathematics for engineers that covers the application of common ratio in engineering.
  • Introduction to Computer Science: A textbook on computer science that covers the application of common ratio in algorithms and data structures.

Additional Resources

  • Online Calculators: Online calculators that can help you calculate the common ratio and sum of an infinite geometric series.
  • Mathematical Software: Mathematical software that can help you analyze and optimize sequences and algorithms.
  • Mathematics Communities: Online communities that can provide you with additional resources and support for learning mathematics.