The Explicit Formula for the Geometric Sequence

Oct 13, 2025 by windailys.com 46 views

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. The explicit formula for a geometric sequence is given by $f(x) = a \cdot r^{x-1}$ , where $a$ is the first term and $r$ is the common ratio. In this article, we will explore the explicit formula for the geometric sequence $-\frac{1}{9}, \frac{1}{3}, -1, 3, -9, \ldots$ and find the common ratio and the recursive formula for this sequence.

The Explicit Formula for the Geometric Sequence

The explicit formula for the geometric sequence $-\frac{1}{9}, \frac{1}{3}, -1, 3, -9, \ldots$ is given by $f(x) = -\frac{1}{9}(-3)^{x-1}$ . This formula tells us that each term in the sequence is obtained by multiplying the previous term by $-3$ .

Finding the Common Ratio

The common ratio of a geometric sequence is the number by which each term is multiplied to obtain the next term. In this case, we can see that each term is obtained by multiplying the previous term by $-3$ . Therefore, the common ratio is $-3$ .

The Recursive Formula for the Geometric Sequence

The recursive formula for a geometric sequence is given by $f(x+1) = r \cdot f(x)$ , where $r$ is the common ratio. In this case, the recursive formula is $f(x+1) = -3 \cdot f(x)$ .

Example of the Recursive Formula

To illustrate the recursive formula, let's consider the first few terms of the sequence:

$f(1) = -\frac{1}{9}$ $f(2) = -3 \cdot f(1) = -3 \cdot -\frac{1}{9} = \frac{1}{3}$ $f(3) = -3 \cdot f(2) = -3 \cdot \frac{1}{3} = -1$ $f(4) = -3 \cdot f(3) = -3 \cdot -1 = 3$ $f(5) = -3 \cdot f(4) = -3 \cdot 3 = -9$

Conclusion

In this article, we have explored the explicit formula for the geometric sequence $-\frac{1}{9}, \frac{1}{3}, -1, 3, -9, \ldots$ and found the common ratio and the recursive formula for this sequence. The common ratio is $-3$ and the recursive formula is $f(x+1) = -3 \cdot f(x)$ .

Common Ratio and Recursive Formula Summary

Option	Common Ratio	Recursive Formula
A	3	$f(x+1) = 3f(x)$
B	-3	$f(x+1) = -3f(x)$

Final Answer

The correct answer is B. The common ratio is $-3$ and the recursive formula is $f(x+1) = -3 \cdot f(x)$ .

Discussion

The explicit formula for a geometric sequence is given by $f(x) = a \cdot r^{x-1}$ , where $a$ is the first term and $r$ is the common ratio. In this case, the explicit formula is $f(x) = -\frac{1}{9}(-3)^{x-1}$ . The common ratio is the number by which each term is multiplied to obtain the next term, and in this case, it is $-3$ . The recursive formula is given by $f(x+1) = r \cdot f(x)$ , and in this case, it is $f(x+1) = -3 \cdot f(x)$ .

Importance of the Common Ratio and Recursive Formula

The common ratio and recursive formula are important concepts in mathematics, particularly in the study of geometric sequences. The common ratio determines the rate at which the terms of the sequence increase or decrease, while the recursive formula provides a way to generate the terms of the sequence iteratively. Understanding the common ratio and recursive formula is essential for working with geometric sequences and for solving problems involving these sequences.

Real-World Applications of Geometric Sequences

Geometric sequences have numerous real-world applications, including:

Finance: Geometric sequences are used to model the growth or decay of investments, such as compound interest.
Biology: Geometric sequences are used to model the growth or decay of populations, such as the growth of bacteria or the decline of a species.
Computer Science: Geometric sequences are used to model the growth or decay of data, such as the growth of a file or the decline of a network.

Conclusion

In conclusion, the explicit formula for the geometric sequence $-\frac{1}{9}, \frac{1}{3}, -1, 3, -9, \ldots$ is $f(x) = -\frac{1}{9}(-3)^{x-1}$ . The common ratio is $-3$ and the recursive formula is $f(x+1) = -3 \cdot f(x)$ . Understanding the common ratio and recursive formula is essential for working with geometric sequences and for solving problems involving these sequences.

Introduction

Geometric sequences are a fundamental concept in mathematics, and they have numerous real-world applications. In our previous article, we explored the explicit formula for the geometric sequence $-\frac{1}{9}, \frac{1}{3}, -1, 3, -9, \ldots$ and found the common ratio and the recursive formula for this sequence. In this article, we will answer some frequently asked questions about geometric sequences and their applications.

Q&A

Q1: What is a geometric sequence?

A1: 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.

Q2: What is the explicit formula for a geometric sequence?

A2: The explicit formula for a geometric sequence is given by $f(x) = a \cdot r^{x-1}$ , where $a$ is the first term and $r$ is the common ratio.

Q3: What is the common ratio of a geometric sequence?

A3: The common ratio of a geometric sequence is the number by which each term is multiplied to obtain the next term.

Q4: What is the recursive formula for a geometric sequence?

A4: The recursive formula for a geometric sequence is given by $f(x+1) = r \cdot f(x)$ , where $r$ is the common ratio.

Q5: What are some real-world applications of geometric sequences?

A5: Geometric sequences have numerous real-world applications, including finance, biology, and computer science. They are used to model the growth or decay of investments, populations, and data.

Q6: How do I find the common ratio of a geometric sequence?

A6: To find the common ratio of a geometric sequence, you can use the formula $r = \frac{f(x+1)}{f(x)}$ , where $f(x)$ is the nth term of the sequence.

Q7: How do I find the recursive formula of a geometric sequence?

A7: To find the recursive formula of a geometric sequence, you can use the formula $f(x+1) = r \cdot f(x)$ , where $r$ is the common ratio.

Q8: What is the difference between an arithmetic sequence and a geometric sequence?

A8: An arithmetic sequence is a type of sequence where each term after the first is found by adding a fixed number to the previous term. A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed number.

Q9: How do I use geometric sequences in finance?

A9: Geometric sequences are used in finance to model the growth or decay of investments, such as compound interest. You can use the formula $f(x) = a \cdot r^{x-1}$ to calculate the future value of an investment.

Q10: How do I use geometric sequences in biology?

A10: Geometric sequences are used in biology to model the growth or decay of populations, such as the growth of bacteria or the decline of a species. You can use the formula $f(x) = a \cdot r^{x-1}$ to calculate the population size at a given time.

Conclusion

In conclusion, geometric sequences are a fundamental concept in mathematics, and they have numerous real-world applications. We hope that this Q&A article has helped to answer some of your questions about geometric sequences and their applications.

Final Tips

Geometric sequences are used to model the growth or decay of investments, populations, and data.
The common ratio is the number by which each term is multiplied to obtain the next term.
The recursive formula is given by $f(x+1) = r \cdot f(x)$ , where $r$ is the common ratio.
Geometric sequences have numerous real-world applications, including finance, biology, and computer science.

Discussion

Geometric sequences are a powerful tool for modeling real-world phenomena. They are used in a wide range of fields, including finance, biology, and computer science. Understanding geometric sequences and their applications can help you to make informed decisions and solve complex problems.

Importance of Geometric Sequences

Geometric sequences are important because they provide a way to model the growth or decay of complex systems. They are used to predict future outcomes and make informed decisions. Understanding geometric sequences and their applications can help you to:

Make informed decisions about investments and financial planning
Model the growth or decay of populations and ecosystems
Predict future outcomes and make informed decisions about complex systems