Which Geometric Series Represents $0.4444 \ldots$ As A Fraction?A. $\frac{1}{4}+\frac{1}{40}+\frac{1}{400}+\frac{1}{4,000}+\ldots$ B. $\frac{1}{40}+\frac{1}{400}+\frac{1}{4,000}+\frac{1}{40,000}+\ldots$ C.

Introduction

In mathematics, a repeating decimal can be represented as a 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. In this article, we will explore how to represent the repeating decimal $0.4444 \ldots$ as a fraction using a geometric series.

Understanding Repeating Decimals

A repeating decimal is a decimal number that has a block of digits that repeats indefinitely. For example, $0.4444 \ldots$ is a repeating decimal where the block of digits $4$ repeats indefinitely. To represent a repeating decimal as a fraction, we need to find the sum of an infinite geometric series.

Representing Repeating Decimals as Geometric Series

A geometric series can be represented as:

$$S = a + ar + ar^2 + ar^3 + \ldots$$

where $a$ is the first term and $r$ is the common ratio. To represent a repeating decimal as a geometric series, we need to find the first term and the common ratio.

Finding the First Term and Common Ratio

Let's consider the repeating decimal $0.4444 \ldots$. We can write it as:

$$0.4444 \ldots = \frac{4}{10} + \frac{4}{10^2} + \frac{4}{10^3} + \ldots$$

We can see that the first term is $\frac{4}{10}$ and the common ratio is $\frac{1}{10}$.

Representing the Geometric Series

Now that we have the first term and the common ratio, we can represent the geometric series as:

$$S = \frac{4}{10} + \frac{4}{10^2} + \frac{4}{10^3} + \ldots$$

This is a geometric series with a first term of $\frac{4}{10}$ and a common ratio of $\frac{1}{10}$.

Finding the Sum of the Geometric Series

The sum of an infinite geometric series can be found using the formula:

$$S = \frac{a}{1 - r}$$

where $a$ is the first term and $r$ is the common ratio. In this case, the first term is $\frac{4}{10}$ and the common ratio is $\frac{1}{10}$.

Calculating the Sum

Now we can calculate the sum of the geometric series:

$$S = \frac{\frac{4}{10}}{1 - \frac{1}{10}}$$

$$S = \frac{\frac{4}{10}}{\frac{9}{10}}$$

$$S = \frac{4}{9}$$

Conclusion

In this article, we have seen how to represent the repeating decimal $0.4444 \ldots$ as a fraction using a geometric series. We have found the first term and the common ratio, represented the geometric series, and calculated the sum of the geometric series. The sum of the geometric series is $\frac{4}{9}$.

Which Geometric Series Represents $0.4444 \ldots$ as a Fraction?

Based on our calculations, we can see that the geometric series:

$$\frac{4}{10} + \frac{4}{10^2} + \frac{4}{10^3} + \ldots$$

represents $0.4444 \ldots$ as a fraction.

Answer

The correct answer is:

A. $\frac{1}{4}+\frac{1}{40}+\frac{1}{400}+\frac{1}{4,000}+\ldots$

This geometric series represents $0.4444 \ldots$ as a fraction.

Discussion

The geometric series:

$$\frac{1}{4}+\frac{1}{40}+\frac{1}{400}+\frac{1}{4,000}+\ldots$$

represents $0.4444 \ldots$ as a fraction. This is because the first term is $\frac{1}{4}$ and the common ratio is $\frac{1}{10}$, which is the same as the repeating decimal $0.4444 \ldots$.

Final Answer

Introduction

In our previous article, we explored how to represent the repeating decimal $0.4444 \ldots$ as a fraction using a geometric series. We found the first term and the common ratio, represented the geometric series, and calculated the sum of the geometric series. In this article, we will answer some frequently asked questions about representing repeating decimals as geometric series.

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 first term and the common ratio of a geometric series?

A: To find the first term and the common ratio of a geometric series, you need to identify the repeating block of digits in the repeating decimal. The first term is the decimal value of the repeating block, and the common ratio is the decimal value of the repeating block divided by 10.

Q: How do I represent a repeating decimal as a geometric series?

A: To represent a repeating decimal as a geometric series, you need to write the repeating decimal as a sum of fractions, where each fraction has a numerator equal to the repeating block of digits and a denominator equal to a power of 10.

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

A: To calculate the sum of a geometric series, you can use the formula:

$$S = \frac{a}{1 - r}$$

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

Q: What is the sum of the geometric series $\frac{1}{4}+\frac{1}{40}+\frac{1}{400}+\frac{1}{4,000}+\ldots$?

A: The sum of the geometric series $\frac{1}{4}+\frac{1}{40}+\frac{1}{400}+\frac{1}{4,000}+\ldots$ is $\frac{4}{9}$.

Q: How do I know which geometric series represents a repeating decimal?

A: To determine which geometric series represents a repeating decimal, you need to identify the repeating block of digits in the repeating decimal and find the corresponding geometric series.

Q: Can I use a geometric series to represent any repeating decimal?

A: Yes, you can use a geometric series to represent any repeating decimal. However, you need to find the first term and the common ratio of the geometric series, and then calculate the sum of the geometric series.

Q: What are some common examples of repeating decimals that can be represented as geometric series?

A: Some common examples of repeating decimals that can be represented as geometric series include:

• $0.1111 \ldots$
• $0.2222 \ldots$
• $0.3333 \ldots$
• $0.4444 \ldots$
• $0.5555 \ldots$

Conclusion

In this article, we have answered some frequently asked questions about representing repeating decimals as geometric series. We have discussed how to find the first term and the common ratio of a geometric series, how to represent a repeating decimal as a geometric series, and how to calculate the sum of a geometric series. We have also provided some common examples of repeating decimals that can be represented as geometric series.

Final Answer

The final answer is A.