Representing Geometric Sequences with Functions

Oct 13, 2025 by windailys.com 73 views

Understanding Geometric Sequences

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 given sequence $$1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots$$ is a classic example of a geometric sequence, where the common ratio is $\frac{1}{3}$ .

Representing Geometric Sequences with Functions

In mathematics, we can represent geometric sequences using functions. The function $y = \left(\frac{1}{3}\right)^n$ represents the given geometric sequence. Here, $n$ is the term number, and $y$ is the corresponding term value.

Domain of the Function

The domain of a function is the set of all possible input values for which the function is defined. In the case of the function $y = \left(\frac{1}{3}\right)^n$ , we need to determine the domain of the function.

Why is the Domain Important?

The domain of a function is crucial in understanding the behavior and properties of the function. In this case, the domain of the function $y = \left(\frac{1}{3}\right)^n$ will help us determine the range of values that the function can take.

Determining the Domain

To determine the domain of the function $y = \left(\frac{1}{3}\right)^n$ , we need to consider the possible values of $n$ . Since the function involves exponentiation, we need to ensure that the exponent is a real number.

Real Numbers as Exponents

In mathematics, we can have real numbers as exponents. For example, $2^{\frac{1}{2}}$ is a valid expression, where $\frac{1}{2}$ is a real number. However, we need to be careful when dealing with negative exponents.

Negative Exponents

A negative exponent is defined as the reciprocal of the positive exponent. For example, $2^{-\frac{1}{2}} = \frac{1}{2^{\frac{1}{2}}}$ . In the case of the function $y = \left(\frac{1}{3}\right)^n$ , we can have negative exponents, but we need to ensure that the base is non-zero.

Non-Zero Base

The base of the function $y = \left(\frac{1}{3}\right)^n$ is $\frac{1}{3}$ , which is a non-zero number. Therefore, we can have negative exponents, and the function is defined for all real numbers.

Domain of the Function

Based on the above discussion, we can conclude that the domain of the function $y = \left(\frac{1}{3}\right)^n$ is all real numbers, including negative numbers. Therefore, the domain of the function is:

A. (0, ∞)

This is incorrect, as the domain includes negative numbers.

Correct Answer

The correct answer is:

A. (-∞, ∞)

This represents all real numbers, including negative numbers.

Conclusion

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions related to the geometric sequence and its representation.

Q: What is a geometric sequence?

A: 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.

Q: What is the common ratio in the given sequence?

A: The common ratio in the given sequence $$1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots$$ is $\frac{1}{3}$ .

Q: How can we represent a geometric sequence using a function?

A: We can represent a geometric sequence using a function of the form $y = ar^n$ , where $a$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: What is the domain of the function $y = \left(\frac{1}{3}\right)^n$ ?

A: The domain of the function $y = \left(\frac{1}{3}\right)^n$ is all real numbers, including negative numbers.

Q: Why is the domain of the function important?

A: The domain of a function is crucial in understanding the behavior and properties of the function. In this case, the domain of the function $y = \left(\frac{1}{3}\right)^n$ will help us determine the range of values that the function can take.

Q: Can we have negative exponents in the function $y = \left(\frac{1}{3}\right)^n$ ?

A: Yes, we can have negative exponents in the function $y = \left(\frac{1}{3}\right)^n$ , but we need to ensure that the base is non-zero.

Q: What is the correct answer for the domain of the function $y = \left(\frac{1}{3}\right)^n$ ?

A: The correct answer for the domain of the function $y = \left(\frac{1}{3}\right)^n$ is A. (-∞, ∞).

Q: How can we determine the range of values that the function $y = \left(\frac{1}{3}\right)^n$ can take?

A: We can determine the range of values that the function $y = \left(\frac{1}{3}\right)^n$ can take by analyzing the behavior of the function as $n$ approaches positive and negative infinity.

Q: What is the behavior of the function $y = \left(\frac{1}{3}\right)^n$ as $n$ approaches positive infinity?

A: As $n$ approaches positive infinity, the function $y = \left(\frac{1}{3}\right)^n$ approaches 0.

Q: What is the behavior of the function $y = \left(\frac{1}{3}\right)^n$ as $n$ approaches negative infinity?

A: As $n$ approaches negative infinity, the function $y = \left(\frac{1}{3}\right)^n$ approaches infinity.

Q: How can we use the behavior of the function $y = \left(\frac{1}{3}\right)^n$ to determine the range of values that it can take?

A: We can use the behavior of the function $y = \left(\frac{1}{3}\right)^n$ to determine the range of values that it can take by analyzing the limits of the function as $n$ approaches positive and negative infinity.

Q: What is the range of values that the function $y = \left(\frac{1}{3}\right)^n$ can take?

A: The range of values that the function $y = \left(\frac{1}{3}\right)^n$ can take is $(0, \infty)$ .

Conclusion

In conclusion, the geometric sequence and its representation are fundamental concepts in mathematics. By understanding the properties of geometric sequences and their representation using functions, we can analyze and solve a wide range of mathematical problems. The domain and range of a function are crucial in understanding its behavior and properties, and we can use these concepts to determine the range of values that a function can take.