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 geometric series is a type of series where each term is obtained by multiplying the previous term by a fixed constant. This constant is called the common ratio. Geometric series have numerous applications in various fields, including finance, engineering, and computer science. In this article, we will explore how to represent repeating decimals as fractions using geometric series.

What is a Repeating Decimal?

A repeating decimal is a decimal number that has a block of digits that repeats indefinitely. For example, 0.4444... is a repeating decimal where the block of digits 4 repeats indefinitely. Repeating decimals can be represented as fractions using geometric series.

Geometric Series Representation of Repeating Decimals

A geometric series can be represented as:

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

where $a$ is the first term and $r$ is the common ratio. To represent a repeating decimal as a fraction using 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... We can write this decimal 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 Repeating Decimal as a Fraction

Using the formula for the sum of an infinite geometric series, we can represent the repeating decimal 0.4444... as a fraction:

$$0.4444\ldots = \frac{\frac{4}{10}}{1 - \frac{1}{10}} = \frac{\frac{4}{10}}{\frac{9}{10}} = \frac{4}{9}$$

Which Geometric Series Represents the Repeating Decimal?

Now that we have represented the repeating decimal 0.4444... as a fraction, we can compare it with the given options.

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. $\frac{1}{9} + \frac{1}{90} + \frac{1}{900} + \frac{1}{9,000} + \ldots$

We can see that option C represents the repeating decimal 0.4444... as a fraction.

Conclusion

In this article, we have explored how to represent repeating decimals as fractions using geometric series. We have found that the repeating decimal 0.4444... can be represented as a fraction using the geometric series $\frac{1}{9} + \frac{1}{90} + \frac{1}{900} + \frac{1}{9,000} + \ldots$. This representation can be useful in various applications, including finance and engineering.

References

• [1] "Geometric Series" by Math Open Reference
• [2] "Repeating Decimals" by Math Is Fun

Further Reading

• [1] "Geometric Series and Repeating Decimals" by Khan Academy
• [2] "Representing Repeating Decimals as Fractions" by Purplemath
  Geometric Series Representation of Repeating Decimals: Q&A =====================================================

Introduction

In our previous article, we explored how to represent repeating decimals as fractions using geometric series. In this article, we will answer some frequently asked questions about geometric series representation of repeating decimals.

Q: What is a geometric series?

A: A geometric series is a type of series where each term is obtained by multiplying the previous term by a fixed constant. This constant is called the common ratio.

Q: How do I find the first term and common ratio of a geometric series?

A: To find the first term and common ratio of a geometric series, you need to identify the pattern of the series. The first term is the first term of the series, and the common ratio is the constant by which each term is multiplied to get the next term.

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

A: To represent a repeating decimal as a fraction using a geometric series, you need to find the first term and common ratio of the series. Then, you can use the formula for the sum of an infinite geometric series to represent the repeating decimal as a fraction.

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: How do I use the formula for the sum of an infinite geometric series to represent a repeating decimal as a fraction?

A: To use the formula for the sum of an infinite geometric series to represent a repeating decimal as a fraction, you need to identify the first term and common ratio of the series. Then, you can plug these values into the formula to get the fraction.

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

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

• 0.4444...
• 0.6666...
• 0.9999...

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

A: To determine which geometric series represents a repeating decimal, you need to compare the series with the repeating decimal. You can do this by converting the repeating decimal to a fraction using a geometric series and comparing it with the given options.

Q: What are some common mistakes to avoid when representing repeating decimals as fractions using geometric series?

A: Some common mistakes to avoid when representing repeating decimals as fractions using geometric series include:

• Not identifying the first term and common ratio of the series
• Not using the correct formula for the sum of an infinite geometric series
• Not converting the repeating decimal to a fraction using a geometric series

Conclusion

In this article, we have answered some frequently asked questions about geometric series representation of repeating decimals. We have provided examples and explanations to help you understand how to represent repeating decimals as fractions using geometric series.

References

• [1] "Geometric Series" by Math Open Reference
• [2] "Repeating Decimals" by Math Is Fun

Further Reading

• [1] "Geometric Series and Repeating Decimals" by Khan Academy
• [2] "Representing Repeating Decimals as Fractions" by Purplemath