Find The 6th Term Of The Series: 1 , − 1 3 , 1 9 , … 1, -\frac{1}{3}, \frac{1}{9}, \ldots 1 , − 3 1 , 9 1 , …

Mar 13, 2025 by ADMIN 114 views

**Finding the 6th Term of a Geometric Series**

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Introduction

In mathematics, a geometric series 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 given series is $1, -\frac{1}{3}, \frac{1}{9}, \ldots$ , and we are asked to find the 6th term of this series. In this article, we will discuss the concept of geometric series, the formula for finding the nth term, and how to apply it to find the 6th term of the given series.

Understanding Geometric Series

A geometric series 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 series is:

$a, ar, ar^2, ar^3, \ldots$

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

Formula for Finding the nth Term

The formula for finding the nth term of a geometric series is:

$a_n = a \cdot r^{n-1}$

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

Finding the Common Ratio

To find the common ratio, we can divide any term by its previous term. In this case, we can divide the second term by the first term:

$\frac{-\frac{1}{3}}{1} = -\frac{1}{3}$

So, the common ratio is $-\frac{1}{3}$ .

Finding the 6th Term

Now that we have the common ratio, we can use the formula to find the 6th term:

$a_6 = 1 \cdot \left(-\frac{1}{3}\right)^{6-1}$

$a_6 = 1 \cdot \left(-\frac{1}{3}\right)^5$

$a_6 = 1 \cdot -\frac{1}{243}$

$a_6 = -\frac{1}{243}$

Therefore, the 6th term of the series is $-\frac{1}{243}$ .

Conclusion

In this article, we discussed the concept of geometric series, the formula for finding the nth term, and how to apply it to find the 6th term of the given series. We found that the common ratio is $-\frac{1}{3}$ and used the formula to find the 6th term, which is $-\frac{1}{243}$ . This demonstrates the importance of understanding geometric series and how to apply mathematical formulas to solve problems.

Real-World Applications

Geometric series have many real-world applications, such as:

Finance: Geometric series are used to calculate compound interest and investment returns.
Physics: Geometric series are used to describe the motion of objects under constant acceleration.
Engineering: Geometric series are used to design and analyze electrical circuits.

Tips and Tricks

When working with geometric series, it's essential to:

Understand the concept of common ratio: The common ratio is the key to finding the nth term of a geometric series.
Use the formula correctly: Make sure to use the correct formula and apply it correctly to find the nth term.
Check your work: Double-check your calculations to ensure accuracy.

Frequently Asked Questions

Q: What is a geometric series?

A: A geometric series 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.

Q: How do I find the common ratio?

A: To find the common ratio, divide any term by its previous term.

Q: How do I find the nth term of a geometric series?

A: Use the formula: $a_n = a \cdot r^{n-1}$ , where $a_n$ is the nth term, $a$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: What are some real-world applications of geometric series?

A: Geometric series have many real-world applications, such as finance, physics, and engineering.

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Introduction

In our previous article, we discussed the concept of geometric series, the formula for finding the nth term, and how to apply it to find the 6th term of a given series. In this article, we will provide a comprehensive Q&A section to address some of the most frequently asked questions about geometric series.

Q&A

Q: What is a geometric series?

A: A geometric series 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.

Q: How do I find the common ratio?

A: To find the common ratio, divide any term by its previous term. For example, if the series is $1, -\frac{1}{3}, \frac{1}{9}, \ldots$ , you can divide the second term by the first term to find the common ratio: $\frac{-\frac{1}{3}}{1} = -\frac{1}{3}$ .

Q: How do I find the nth term of a geometric series?

A: Use the formula: $a_n = a \cdot r^{n-1}$ , where $a_n$ is the nth term, $a$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: What is the difference between an arithmetic series and a geometric series?

A: An arithmetic series is a sequence of numbers where each term after the first is found by adding a fixed number called the common difference. A geometric series, on the other hand, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed number called the common ratio.

Q: How do I determine if a series is geometric or arithmetic?

A: To determine if a series is geometric or arithmetic, look for the pattern of the terms. If the terms are obtained by multiplying the previous term by a fixed number, it's a geometric series. If the terms are obtained by adding a fixed number, it's an arithmetic series.

Q: Can I have a negative common ratio?

A: Yes, you can have a negative common ratio. In fact, the common ratio can be any real number, positive or negative.

Q: Can I have a common ratio of 0?

A: No, you cannot have a common ratio of 0. If the common ratio is 0, the series will be a constant sequence, and the formula for the nth term will not be applicable.

Q: Can I have a common ratio of 1?

A: Yes, you can have a common ratio of 1. In this case, the series will be a constant sequence, and the formula for the nth term will be $a_n = a$ .

Q: How do I find the sum of a geometric series?

A: The sum of a geometric series can be found using the formula: $S_n = \frac{a(1-r^n)}{1-r}$ , 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 formula for the sum of an infinite geometric series?

A: The formula for the sum of an infinite geometric series is: $S = \frac{a}{1-r}$ , where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio.

Conclusion

In this article, we provided a comprehensive Q&A section to address some of the most frequently asked questions about geometric series. We hope that this article has been helpful in clarifying any doubts you may have had about geometric series.

Real-World Applications

Geometric series have many real-world applications, such as:

Finance: Geometric series are used to calculate compound interest and investment returns.
Physics: Geometric series are used to describe the motion of objects under constant acceleration.
Engineering: Geometric series are used to design and analyze electrical circuits.

Tips and Tricks

When working with geometric series, it's essential to:

Understand the concept of common ratio: The common ratio is the key to finding the nth term of a geometric series.
Use the formula correctly: Make sure to use the correct formula and apply it correctly to find the nth term.
Check your work: Double-check your calculations to ensure accuracy.

Frequently Asked Questions (FAQs)

Q: What is a geometric series?

A: A geometric series 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.

Q: How do I find the common ratio?

A: To find the common ratio, divide any term by its previous term.

Q: How do I find the nth term of a geometric series?

A: Use the formula: $a_n = a \cdot r^{n-1}$ , where $a_n$ is the nth term, $a$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: What are some real-world applications of geometric series?

A: Geometric series have many real-world applications, such as finance, physics, and engineering.