The Given Geometric Sequence

Oct 13, 2025 by windailys.com 63 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. In this article, we will explore the geometric sequence ${$ \left{\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81} \ldots\right}$}$ and find the explicit formula that describes it.

What is a Geometric Sequence?

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric sequence is:

a, ar, ar^2, ar^3, ...

where a is the first term and r is the common ratio.

The Given Geometric Sequence

The given geometric sequence is ${$ \left{\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81} \ldots\right}$}$. We can see that the first term is ${$ \frac{1}{3}$}$ and the common ratio is ${$ \frac{1}{3}$}$ as well.

Finding the Explicit Formula

The explicit formula for a geometric sequence is given by:

a_n = a * r^(n-1)

where a_n is the nth term of the sequence, a is the first term, r is the common ratio, and n is the term number.

Option A: a_n = \frac{1}{3} + (n-1)\left(\frac{1}{3}\right)

Let's analyze option A: a_n = \frac{1}{3} + (n-1)\left(\frac{1}{3}\right). We can see that the first term is ${$ \frac{1}{3}$}$ and the common ratio is ${$ \frac{1}{3}$}$ as well. However, the formula is not in the standard form of a geometric sequence.

**Option B: a_n = \frac{1}{3} * \left(\frac{1}{3}\right)^{n-1}**

Let's analyze option B: a_n = \frac{1}{3} * \left(\frac{1}{3}\right)^{n-1}. We can see that the first term is ${$ \frac{1}{3}$}$ and the common ratio is ${$ \frac{1}{3}$}$ as well. This formula is in the standard form of a geometric sequence.

Conclusion

Based on our analysis, we can conclude that the explicit formula that describes the geometric sequence ${$ \left{\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81} \ldots\right}$}$ is:

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

This formula is in the standard form of a geometric sequence and accurately describes the given sequence.

Why is this Formula Important?

The explicit formula for a geometric sequence is important because it allows us to find any term of the sequence without having to calculate all the previous terms. This is particularly useful when working with large sequences or when we need to find a specific term quickly.

Real-World Applications

Geometric sequences have many real-world applications, including:

Finance: Geometric sequences are used to calculate compound interest and investment returns.
Science: Geometric sequences are used to model population growth and decay.
Engineering: Geometric sequences are used to design and analyze electrical circuits.

Conclusion

In conclusion, the explicit formula that describes the geometric sequence ${$ \left{\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81} \ldots\right}$}$ is:

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

Introduction

Geometric sequences are a fundamental concept in mathematics, and understanding them is crucial for solving problems in various fields, including finance, science, and engineering. In this article, we will provide a comprehensive Q&A guide to help you understand geometric sequences and their applications.

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 Formula for a Geometric Sequence?

A: The formula for a geometric sequence is:

a_n = a * r^(n-1)

where a_n is the nth term of the sequence, a is the first term, r is the common ratio, and n is the term number.

Q: How Do I Find the Common Ratio?

A: To find the common ratio, you can divide any term by its previous term. For example, if the sequence is ${$ \left{2, 6, 18, 54 \ldots\right}$}$, you can divide the second term (6) by the first term (2) to get the common ratio (3).

Q: How Do I Find the nth Term of a Geometric Sequence?

A: To find the nth term of a geometric sequence, you can use the formula:

a_n = a * r^(n-1)

where a_n is the nth term of the sequence, a is the first term, r is the common ratio, and n is the term number.

Q: What is the Sum of a Geometric Sequence?

A: The sum of a geometric sequence is given by the formula:

S_n = a * \frac{1 - r^n}{1 - r}

where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.

Q: What is the Product of a Geometric Sequence?

A: The product of a geometric sequence is given by the formula:

P_n = a * r^0 * r^1 * r^2 * ... * r^(n-1)

where P_n is the product of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.

Q: How Do I Use Geometric Sequences in Real-World Applications?

A: Geometric sequences have many real-world applications, including:

Finance: Geometric sequences are used to calculate compound interest and investment returns.
Science: Geometric sequences are used to model population growth and decay.
Engineering: Geometric sequences are used to design and analyze electrical circuits.

Q: What are Some Common Mistakes to Avoid When Working with Geometric Sequences?

A: Some common mistakes to avoid when working with geometric sequences include:

Not checking for a common ratio: Make sure to check if the sequence has a common ratio before using the formula.
Not using the correct formula: Use the correct formula for the sum or product of a geometric sequence.
Not considering the number of terms: Make sure to consider the number of terms when calculating the sum or product of a geometric sequence.

Conclusion

In conclusion, geometric sequences are a fundamental concept in mathematics, and understanding them is crucial for solving problems in various fields. By following the Q&A guide provided in this article, you will be able to understand geometric sequences and their applications, and you will be able to avoid common mistakes when working with them.