Understanding the Infinite Geometric Sequence

Oct 13, 2025 by miodeya.com 104 views

Introduction

In mathematics, an infinite geometric series is a series of the form $a + ar + ar^2 + ar^3 + \ldots$ , where $a$ is the first term and $r$ is the common ratio. The sum of an infinite geometric series can be found using the formula $S = \frac{a}{1 - r}$ , but only if the absolute value of $r$ is less than 1. In this article, we will explore how to find the sum of an infinite geometric series related to the infinite geometric sequence $(1,521; 1,369; 1,232; \ldots)$ .

Understanding the Infinite Geometric Sequence

The given infinite geometric sequence is $(1,521; 1,369; 1,232; \ldots)$ . To find the sum of the infinite geometric series related to this sequence, we need to find the common ratio $r$ . The common ratio is the ratio of any term to its previous term. In this case, we can find the common ratio by dividing the second term by the first term:

$r = \frac{1369}{1521} = 0.8985$

Rounding the Value of $r$

The value of $r$ needs to be rounded to the nearest hundredth. Therefore, we round $0.8985$ to $0.90$ .

Finding the Sum of the Infinite Geometric Series

Now that we have the value of $r$ , we can find the sum of the infinite geometric series using the formula $S = \frac{a}{1 - r}$ . The first term $a$ is 1521. Plugging in the values, we get:

$S = \frac{1521}{1 - 0.90} = \frac{1521}{0.10} = 15210$

Conclusion

In this article, we found the sum of the infinite geometric series related to the infinite geometric sequence $(1,521; 1,369; 1,232; \ldots)$ . We first found the common ratio $r$ by dividing the second term by the first term. We then rounded the value of $r$ to the nearest hundredth. Finally, we used the formula $S = \frac{a}{1 - r}$ to find the sum of the infinite geometric series.

The Formula for the Sum of an Infinite Geometric Series

The formula for the sum of an infinite geometric series is $S = \frac{a}{1 - r}$ , where $a$ is the first term and $r$ is the common ratio. This formula is only valid if the absolute value of $r$ is less than 1.

Example of Finding the Sum of an Infinite Geometric Series

Let's say we have an infinite geometric series with a first term of 2 and a common ratio of 0.5. We can find the sum of this series using the formula $S = \frac{a}{1 - r}$ :

$S = \frac{2}{1 - 0.5} = \frac{2}{0.5} = 4$

Properties of Infinite Geometric Series

Infinite geometric series have several properties that make them useful in mathematics. Some of these properties include:

The sum of an infinite geometric series is only defined if the absolute value of the common ratio is less than 1.
The sum of an infinite geometric series can be found using the formula $S = \frac{a}{1 - r}$ .
The formula for the sum of an infinite geometric series is only valid if the absolute value of the common ratio is less than 1.

Real-World Applications of Infinite Geometric Series

Infinite geometric series have several real-world applications. Some of these applications include:

Finance: Infinite geometric series can be used to model the growth of investments over time.
Physics: Infinite geometric series can be used to model the decay of radioactive materials over time.
Engineering: Infinite geometric series can be used to model the behavior of electrical circuits.

Conclusion

$In this article, we found the sum of the infinite geometric series related to the infinite geometric sequence $(1,521; 1,369; 1,232; \ldots)$. We first found the common ratio $r$ by dividing the second term by the first term. We then rounded the value of $r$ to the nearest hundredth. Finally, we used the formula $S = \frac{a}{1 - r}$ to find the sum of the infinite geometric series. We also discussed the formula for the sum of an infinite geometric series and its properties.<br/> **Frequently Asked Questions (FAQs) About Infinite Geometric Series**$

Q: What is an infinite geometric series?

A: An infinite geometric series is a series of the form $a + ar + ar^2 + ar^3 + \ldots$ , where $a$ is the first term and $r$ is the common ratio.

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

Q: When is the sum of an infinite geometric series defined?

A: The sum of an infinite geometric series is only defined if the absolute value of the common ratio is less than 1.

Q: How do I find the common ratio of an infinite geometric series?

A: To find the common ratio of an infinite geometric series, you can divide any term by its previous term.

Q: What is the significance of the common ratio in an infinite geometric series?

A: The common ratio determines whether the series converges or diverges. If the absolute value of the common ratio is less than 1, the series converges. If the absolute value of the common ratio is greater than or equal to 1, the series diverges.

Q: Can I use the formula for the sum of an infinite geometric series if the common ratio is greater than or equal to 1?

A: No, you cannot use the formula for the sum of an infinite geometric series if the common ratio is greater than or equal to 1. In this case, the series diverges, and the formula is not applicable.

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

A: Infinite geometric series have several real-world applications, including:

Finance: Infinite geometric series can be used to model the growth of investments over time.
Physics: Infinite geometric series can be used to model the decay of radioactive materials over time.
Engineering: Infinite geometric series can be used to model the behavior of electrical circuits.

Q: How do I round the value of the common ratio to the nearest hundredth?

A: To round the value of the common ratio to the nearest hundredth, you can look at the thousandth place. If the thousandth place is 5 or greater, you round up. If the thousandth place is less than 5, you round down.

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

A: A finite geometric series is a series with a finite number of terms, while an infinite geometric series is a series with an infinite number of terms.

Q: Can I use the formula for the sum of a finite geometric series if the series has an infinite number of terms?

A: No, you cannot use the formula for the sum of a finite geometric series if the series has an infinite number of terms. In this case, you need to use the formula for the sum of an infinite geometric series.

Q: What are some common mistakes to avoid when working with infinite geometric series?

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

Not checking if the absolute value of the common ratio is less than 1
Not rounding the value of the common ratio to the nearest hundredth
Not using the correct formula for the sum of an infinite geometric series

Conclusion

In this article, we answered some frequently asked questions about infinite geometric series. We discussed the formula for the sum of an infinite geometric series, the significance of the common ratio, and some real-world applications of infinite geometric series. We also provided some tips on how to avoid common mistakes when working with infinite geometric series.