The Explicit Rule For A Sequence Is Given:$\[ A_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1} \\]Enter The Recursive Rule For The Geometric Sequence:$\[ A_1 = \square ; \quad A_n = \square \\]

Introduction

In mathematics, a sequence is a series of numbers in a specific order. A geometric sequence is a type of sequence where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. The explicit rule for a geometric sequence is given as $a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$. In this article, we will explore the recursive rule for the geometric sequence and provide a detailed analysis of the given explicit rule.

Understanding the Explicit Rule

The explicit rule for a geometric sequence is given as $a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$. This rule provides a direct formula for calculating the nth term of the sequence. To understand this rule, let's break it down into its components.

• The term $\frac{1}{2}$ is the first term of the sequence, denoted as $a_1$.
• The term $\left(\frac{1}{3}\right)^{n-1}$ represents the common ratio, denoted as $r$. In this case, the common ratio is $\frac{1}{3}$.

Deriving the Recursive Rule

The recursive rule for a geometric sequence is given as $a_n = a_{n-1} \cdot r$, where $a_n$ is the nth term of the sequence, $a_{n-1}$ is the previous term, and $r$ is the common ratio. To derive the recursive rule from the explicit rule, we can start by substituting the explicit rule into the recursive rule.

Step 1: Substitute the Explicit Rule into the Recursive Rule

Let's substitute the explicit rule $a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$ into the recursive rule $a_n = a_{n-1} \cdot r$.

Step 2: Simplify the Recursive Rule

By substituting the explicit rule into the recursive rule, we get:

$a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1} = \frac{1}{2}\left(\frac{1}{3}\right)^{n-2} \cdot \frac{1}{3}$

Step 3: Identify the Recursive Rule

By simplifying the recursive rule, we can identify the recursive rule for the geometric sequence as:

$a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-2} \cdot \frac{1}{3} = a_{n-1} \cdot \frac{1}{3}$

Conclusion

In this article, we have derived the recursive rule for the geometric sequence from the explicit rule. The recursive rule is given as $a_n = a_{n-1} \cdot \frac{1}{3}$, where $a_n$ is the nth term of the sequence, $a_{n-1}$ is the previous term, and $\frac{1}{3}$ is the common ratio. This recursive rule provides a direct formula for calculating the nth term of the sequence and can be used to analyze and understand the properties of the geometric sequence.

Geometric Sequence Analysis

A geometric sequence is a type of sequence where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. The explicit rule for a geometric sequence is given as $a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$. In this section, we will analyze the properties of the geometric sequence and provide a detailed explanation of the recursive rule.

Properties of the Geometric Sequence

The geometric sequence has several properties that can be analyzed using the recursive rule. Some of the key properties include:

• Common Ratio: The common ratio is the fixed constant that is multiplied by the previous term to obtain the next term. In this case, the common ratio is $\frac{1}{3}$.
• First Term: The first term of the sequence is denoted as $a_1$ and is given as $\frac{1}{2}$.
• Nth Term: The nth term of the sequence is denoted as $a_n$ and can be calculated using the recursive rule.

Recursive Rule Analysis

The recursive rule for the geometric sequence is given as $a_n = a_{n-1} \cdot \frac{1}{3}$. This rule provides a direct formula for calculating the nth term of the sequence and can be used to analyze and understand the properties of the geometric sequence.

• Recursive Formula: The recursive formula is a direct formula for calculating the nth term of the sequence. It is given as $a_n = a_{n-1} \cdot \frac{1}{3}$.
• Common Ratio: The common ratio is the fixed constant that is multiplied by the previous term to obtain the next term. In this case, the common ratio is $\frac{1}{3}$.
• First Term: The first term of the sequence is denoted as $a_1$ and is given as $\frac{1}{2}$.

Conclusion

In this article, we have analyzed the properties of the geometric sequence and provided a detailed explanation of the recursive rule. The recursive rule is given as $a_n = a_{n-1} \cdot \frac{1}{3}$, where $a_n$ is the nth term of the sequence, $a_{n-1}$ is the previous term, and $\frac{1}{3}$ is the common ratio. This recursive rule provides a direct formula for calculating the nth term of the sequence and can be used to analyze and understand the properties of the geometric sequence.

Geometric Sequence Formula

The geometric sequence formula is given as $a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$. This formula provides a direct formula for calculating the nth term of the sequence and can be used to analyze and understand the properties of the geometric sequence.

Geometric Sequence Formula Derivation

The geometric sequence formula can be derived using the recursive rule. To derive the formula, we can start by substituting the recursive rule into the formula.

Step 1: Substitute the Recursive Rule into the Formula

Let's substitute the recursive rule $a_n = a_{n-1} \cdot \frac{1}{3}$ into the formula.

Step 2: Simplify the Formula

By substituting the recursive rule into the formula, we get:

$a_n = a_{n-1} \cdot \frac{1}{3} = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$

Step 3: Identify the Geometric Sequence Formula

By simplifying the formula, we can identify the geometric sequence formula as:

$a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$

Conclusion

In this article, we have derived the geometric sequence formula from the recursive rule. The geometric sequence formula is given as $a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$. This formula provides a direct formula for calculating the nth term of the sequence and can be used to analyze and understand the properties of the geometric sequence.

Geometric Sequence Applications

The geometric sequence has several applications in mathematics and other fields. Some of the key applications include:

• Finance: Geometric sequences are used to calculate compound interest and other financial calculations.
• Biology: Geometric sequences are used to model population growth and other biological processes.
• Computer Science: Geometric sequences are used to model algorithms and other computational processes.

Conclusion

In this article, we have analyzed the properties of the geometric sequence and provided a detailed explanation of the recursive rule. The recursive rule is given as $a_n = a_{n-1} \cdot \frac{1}{3}$, where $a_n$ is the nth term of the sequence, $a_{n-1}$ is the previous term, and $\frac{1}{3}$ is the common ratio. This recursive rule provides a direct formula for calculating the nth term of the sequence and can be used to analyze and understand the properties of the geometric sequence.

Introduction

In this article, we will provide a comprehensive Q&A section on geometric sequences. Geometric sequences are a type of sequence where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. The explicit rule for a geometric sequence is given as $a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$. We will answer some of the most frequently asked questions about geometric sequences and provide a detailed explanation of the concepts.

Q1: What is a geometric sequence?

A1: A geometric sequence is a type of sequence where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio.

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

A2: The explicit rule for a geometric sequence is given as $a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$.

Q3: What is the recursive rule for a geometric sequence?

A3: The recursive rule for a geometric sequence is given as $a_n = a_{n-1} \cdot \frac{1}{3}$.

Q4: What is the common ratio in a geometric sequence?

A4: The common ratio in a geometric sequence is the fixed constant that is multiplied by the previous term to obtain the next term.

Q5: How do I calculate the nth term of a geometric sequence?

A5: To calculate the nth term of a geometric sequence, you can use the explicit rule $a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$ or the recursive rule $a_n = a_{n-1} \cdot \frac{1}{3}$.

Q6: What are some applications of geometric sequences?

A6: Geometric sequences have several applications in mathematics and other fields, including finance, biology, and computer science.

Q7: How do I determine the common ratio of a geometric sequence?

A7: To determine the common ratio of a geometric sequence, you can use the formula $r = \frac{a_n}{a_{n-1}}$, where $r$ is the common ratio and $a_n$ and $a_{n-1}$ are the nth and (n-1)th terms of the sequence.

Q8: What is the formula for the sum of a geometric sequence?

A8: The formula for the sum of a geometric sequence is given as $S_n = \frac{a_1(1-r^n)}{1-r}$, where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q9: How do I calculate the sum of a geometric sequence?

A9: To calculate the sum of a geometric sequence, you can use the formula $S_n = \frac{a_1(1-r^n)}{1-r}$.

Q10: What are some real-world examples of geometric sequences?

A10: Some real-world examples of geometric sequences include population growth, compound interest, and the spread of a disease.

Conclusion

In this article, we have provided a comprehensive Q&A section on geometric sequences. We have answered some of the most frequently asked questions about geometric sequences and provided a detailed explanation of the concepts. Geometric sequences are a type of sequence where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. The explicit rule for a geometric sequence is given as $a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$. We hope that this article has provided a helpful resource for understanding geometric sequences.

Geometric Sequence Formula Derivation

The geometric sequence formula can be derived using the recursive rule. To derive the formula, we can start by substituting the recursive rule into the formula.

Step 1: Substitute the Recursive Rule into the Formula

Let's substitute the recursive rule $a_n = a_{n-1} \cdot \frac{1}{3}$ into the formula.

Step 2: Simplify the Formula

By substituting the recursive rule into the formula, we get:

$a_n = a_{n-1} \cdot \frac{1}{3} = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$

Step 3: Identify the Geometric Sequence Formula

By simplifying the formula, we can identify the geometric sequence formula as:

$a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$

Conclusion

In this article, we have derived the geometric sequence formula from the recursive rule. The geometric sequence formula is given as $a_n = \frac{1}{2}\left(\frac{1}{3}\right)^{n-1}$. This formula provides a direct formula for calculating the nth term of the sequence and can be used to analyze and understand the properties of the geometric sequence.

Geometric Sequence Applications

The geometric sequence has several applications in mathematics and other fields. Some of the key applications include:

• Finance: Geometric sequences are used to calculate compound interest and other financial calculations.
• Biology: Geometric sequences are used to model population growth and other biological processes.
• Computer Science: Geometric sequences are used to model algorithms and other computational processes.

Conclusion

In this article, we have analyzed the properties of the geometric sequence and provided a detailed explanation of the recursive rule. The recursive rule is given as $a_n = a_{n-1} \cdot \frac{1}{3}$, where $a_n$ is the nth term of the sequence, $a_{n-1}$ is the previous term, and $\frac{1}{3}$ is the common ratio. This recursive rule provides a direct formula for calculating the nth term of the sequence and can be used to analyze and understand the properties of the geometric sequence.