Ratio of consecutive terms

Oct 13, 2025 by pinkandtrash.com 29 views

What is the Common Ratio of the Sequence $3, 21, 147, \ldots$ ?

In mathematics, a sequence is a series of numbers in a specific order. The common ratio of a sequence is the ratio of any two consecutive terms in the sequence. It is an essential concept in understanding the behavior and properties of sequences. In this article, we will explore the common ratio of the sequence $3, 21, 147, \ldots$ .

The given sequence is $3, 21, 147, \ldots$ . To find the common ratio, we need to examine the relationship between consecutive terms. Let's calculate the ratio of the second term to the first term, and the ratio of the third term to the second term.

The ratio of the second term to the first term is $\frac{21}{3} = 7$ .
The ratio of the third term to the second term is $\frac{147}{21} = 7$ .

As we can see, both ratios are equal to $7$ . This indicates that the common ratio of the sequence is $7$ .

The common ratio has several important properties that are essential in understanding sequences. Some of these properties include:

Multiplicative property: The common ratio is the same for all consecutive terms in the sequence.
Constant ratio: The common ratio is a constant value that does not change from one term to the next.
Ratio of consecutive terms: The common ratio is the ratio of any two consecutive terms in the sequence.

Sequences with common ratios are abundant in mathematics. Here are a few examples:

Geometric sequence: A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a fixed constant. The common ratio of a geometric sequence is the fixed constant.
Arithmetic sequence: An arithmetic sequence is a sequence in which each term is obtained by adding a fixed constant to the previous term. The common ratio of an arithmetic sequence is the fixed constant divided by the number of terms.
Fibonacci sequence: The Fibonacci sequence is a sequence in which each term is the sum of the two preceding terms. The common ratio of the Fibonacci sequence is the golden ratio, which is approximately equal to $1.618$ .

Common ratios have numerous real-world applications in various fields, including:

Finance: Common ratios are used in finance to calculate interest rates, investment returns, and other financial metrics.
Science: Common ratios are used in science to model population growth, chemical reactions, and other phenomena.
Engineering: Common ratios are used in engineering to design and optimize systems, such as electrical circuits and mechanical systems.

In conclusion, the common ratio of the sequence $3, 21, 147, \ldots$ is $7$ . The common ratio is an essential concept in understanding sequences and has numerous real-world applications. By understanding the properties and behavior of common ratios, we can better analyze and model complex phenomena in various fields.

Wikipedia: Common ratio. Retrieved from https://en.wikipedia.org/wiki/Common_ratio
Mathworld: Common ratio. Retrieved from https://mathworld.wolfram.com/CommonRatio.html
Khan Academy: Sequences and series. Retrieved from https://www.khanacademy.org/math/sequences-series

For further reading on sequences and common ratios, we recommend the following resources:

"Sequences and Series" by Khan Academy: This article provides an in-depth introduction to sequences and series, including common ratios.
"Common Ratio" by Mathworld: This article provides a comprehensive overview of common ratios, including their properties and applications.
"Sequences and Series" by Wikipedia: This article provides a detailed introduction to sequences and series, including common ratios.
Frequently Asked Questions (FAQs) About Common Ratios

In our previous article, we explored the concept of common ratios and their properties. In this article, we will answer some frequently asked questions (FAQs) about common ratios to provide a deeper understanding of this essential mathematical concept.

A: A common ratio is the ratio of any two consecutive terms in a sequence. It is a constant value that does not change from one term to the next.

A: To find the common ratio of a sequence, you can divide any term by its previous term. For example, if the sequence is $3, 21, 147, \ldots$ , you can divide the second term ( $21$ ) by the first term ( $3$ ) to get the common ratio, which is $7$ .

A: The properties of a common ratio include:

Multiplicative property: The common ratio is the same for all consecutive terms in the sequence.
Constant ratio: The common ratio is a constant value that does not change from one term to the next.
Ratio of consecutive terms: The common ratio is the ratio of any two consecutive terms in the sequence.

A: Some examples of sequences with common ratios include:

Geometric sequence: A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a fixed constant. The common ratio of a geometric sequence is the fixed constant.
Arithmetic sequence: An arithmetic sequence is a sequence in which each term is obtained by adding a fixed constant to the previous term. The common ratio of an arithmetic sequence is the fixed constant divided by the number of terms.
Fibonacci sequence: The Fibonacci sequence is a sequence in which each term is the sum of the two preceding terms. The common ratio of the Fibonacci sequence is the golden ratio, which is approximately equal to $1.618$ .

A: Common ratios have numerous real-world applications in various fields, including:

Finance: Common ratios are used in finance to calculate interest rates, investment returns, and other financial metrics.
Science: Common ratios are used in science to model population growth, chemical reactions, and other phenomena.
Engineering: Common ratios are used in engineering to design and optimize systems, such as electrical circuits and mechanical systems.

A: Yes, you can use a calculator to find the common ratio of a sequence. Simply divide any term by its previous term to get the common ratio.

A: If you have a sequence with a common ratio of $0$ , it means that the sequence is not geometric. In this case, you may need to use other methods to analyze the sequence.

A: Yes, you can use the common ratio to predict future terms in a sequence. Simply multiply the last term by the common ratio to get the next term.

In conclusion, common ratios are an essential concept in mathematics that have numerous real-world applications. By understanding the properties and behavior of common ratios, we can better analyze and model complex phenomena in various fields. We hope that this FAQ article has provided a deeper understanding of common ratios and their applications.

Wikipedia: Common ratio. Retrieved from https://en.wikipedia.org/wiki/Common_ratio
Mathworld: Common ratio. Retrieved from https://mathworld.wolfram.com/CommonRatio.html
Khan Academy: Sequences and series. Retrieved from https://www.khanacademy.org/math/sequences-series

For further reading on sequences and common ratios, we recommend the following resources:

"Sequences and Series" by Khan Academy: This article provides an in-depth introduction to sequences and series, including common ratios.
"Common Ratio" by Mathworld: This article provides a comprehensive overview of common ratios, including their properties and applications.
"Sequences and Series" by Wikipedia: This article provides a detailed introduction to sequences and series, including common ratios.