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 . \[ B. \[ B . \[ \frac{1}{40} + \frac{1}{400} + \frac{1}{4,000} + \frac{1}{40,000} +

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. On the other hand, a repeating decimal is a decimal number that goes on indefinitely in a predictable pattern. In this article, we will explore the connection between geometric series and repeating decimals, and use this connection to find the fraction that represents the repeating decimal $0.4444\ldots$.

What is a Geometric Series?

A geometric series is a sequence of numbers that can be written in the form:

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

where $a$ is the first term and $r$ is the common ratio. The sum of a geometric series can be found using the formula:

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

when $|r| < 1$. This formula is known as the formula for the sum of an infinite geometric series.

What is a Repeating Decimal?

A repeating decimal is a decimal number that goes on indefinitely in a predictable pattern. For example, the decimal number $0.4444\ldots$ is a repeating decimal because the digit $4$ repeats indefinitely. Repeating decimals can be written in the form:

$$0.\overline{a_1a_2a_3\ldots}$$

where $a_1a_2a_3\ldots$ is the repeating pattern.

Converting Repeating Decimals to Fractions

To convert a repeating decimal to a fraction, we can use the following steps:

1. Let $x$ be the repeating decimal.
2. Multiply $x$ by a power of $10$ that is equal to the number of digits in the repeating pattern.
3. Subtract the original repeating decimal from the result.
4. The result is a fraction that represents the repeating decimal.

Example: Converting $0.4444\ldots$ to a Fraction

Let $x = 0.4444\ldots$. We can convert $x$ to a fraction using the following steps:

1. Let $x = 0.4444\ldots$.
2. Multiply $x$ by $10$ to get $10x = 4.4444\ldots$.
3. Subtract the original repeating decimal from the result to get $10x - x = 4.4444\ldots - 0.4444\ldots = 4$.
4. The result is a fraction that represents the repeating decimal: $x = \frac{4}{9}$.

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

Now that we have converted the repeating decimal $0.4444\ldots$ to a fraction, we can use this fraction to find the geometric series that represents it. The fraction $\frac{4}{9}$ can be written as a geometric series as follows:

$$\frac{4}{9} = \frac{4}{10} + \frac{4}{100} + \frac{4}{1000} + \frac{4}{10000} + \ldots$$

This geometric series has a first term of $\frac{4}{10}$ and a common ratio of $\frac{1}{10}$. Therefore, the geometric series that represents the repeating decimal $0.4444\ldots$ as a fraction is:

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

This geometric series has a first term of $\frac{1}{4}$ and a common ratio of $\frac{1}{10}$. Therefore, it represents the repeating decimal $0.4444\ldots$ as a fraction.

Conclusion

In this article, we have explored the connection between geometric series and repeating decimals. We have used this connection to convert the repeating decimal $0.4444\ldots$ to a fraction, and have found the geometric series that represents it. The geometric series that represents the repeating decimal $0.4444\ldots$ as a fraction is:

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

Discussion

The connection between geometric series and repeating decimals is a powerful tool for converting repeating decimals to fractions. By using this connection, we can find the fraction that represents a repeating decimal, and can use this fraction to solve a wide range of mathematical problems.

References

• [1] "Geometric Series" by Math Open Reference. Retrieved from https://www.mathopenref.com/seriesgeometric.html
• [2] "Repeating Decimals" by Math Is Fun. Retrieved from https://www.mathisfun.com/numbers/repeating-decimals.html

Additional Resources

• [1] "Geometric Series and Repeating Decimals" by Khan Academy. Retrieved from https://www.khanacademy.org/math/pre-algebra/pre-algebra-decimals/series-and-decimals/v/geometric-series-and-repeating-decimals
• [2] "Converting Repeating Decimals to Fractions" by Purplemath. Retrieved from https://www.purplemath.com/modules/decimals.htm
  Geometric Series and Repeating Decimals: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the connection between geometric series and repeating decimals. We used this connection to convert the repeating decimal $0.4444\ldots$ to a fraction, and found the geometric series that represents it. In this article, we will answer some common questions about geometric series and repeating decimals.

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: What is a repeating decimal?

A: A repeating decimal is a decimal number that goes on indefinitely in a predictable pattern.

Q: How do I convert a repeating decimal to a fraction?

A: To convert a repeating decimal to a fraction, you can use the following steps:

1. Let $x$ be the repeating decimal.
2. Multiply $x$ by a power of $10$ that is equal to the number of digits in the repeating pattern.
3. Subtract the original repeating decimal from the result.
4. The result is a fraction that represents the repeating decimal.

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:

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

when $|r| < 1$, where $a$ is the first term and $r$ is the common ratio.

Q: How do I find the geometric series that represents a repeating decimal?

A: To find the geometric series that represents a repeating decimal, you can use the following steps:

1. Convert the repeating decimal to a fraction using the steps above.
2. Write the fraction as a geometric series by finding the first term and the common ratio.
3. The geometric series that represents the repeating decimal is the one with the same first term and common ratio.

Q: What is the difference between a geometric series and a repeating decimal?

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. A repeating decimal is a decimal number that goes on indefinitely in a predictable pattern. While a geometric series can be used to represent a repeating decimal, they are not the same thing.

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, the geometric series may not be unique, and there may be multiple geometric series that represent the same repeating decimal.

Q: How do I use a geometric series to solve a problem?

A: To use a geometric series to solve a problem, you can follow these steps:

1. Identify the repeating decimal in the problem.
2. Convert the repeating decimal to a fraction using the steps above.
3. Write the fraction as a geometric series by finding the first term and the common ratio.
4. Use the formula for the sum of an infinite geometric series to find the solution to the problem.

Conclusion

In this article, we have answered some common questions about geometric series and repeating decimals. We have also provided a step-by-step guide on how to use a geometric series to represent a repeating decimal and solve a problem. By following these steps, you can use geometric series to solve a wide range of mathematical problems.

Discussion

The connection between geometric series and repeating decimals is a powerful tool for solving mathematical problems. By using this connection, you can convert repeating decimals to fractions and use geometric series to solve a wide range of problems.

References

• [1] "Geometric Series" by Math Open Reference. Retrieved from https://www.mathopenref.com/seriesgeometric.html
• [2] "Repeating Decimals" by Math Is Fun. Retrieved from https://www.mathisfun.com/numbers/repeating-decimals.html

Additional Resources

• [1] "Geometric Series and Repeating Decimals" by Khan Academy. Retrieved from https://www.khanacademy.org/math/pre-algebra/pre-algebra-decimals/series-and-decimals/v/geometric-series-and-repeating-decimals
• [2] "Converting Repeating Decimals to Fractions" by Purplemath. Retrieved from https://www.purplemath.com/modules/decimals.htm