Question

which geometric series represents 0.4444... as a fraction?
○ \\(\frac{1}{4} + \frac{1}{40} + \frac{1}{400} + \frac{1}{4,000} + \dots\\)
○ \\(\frac{1}{40} + \frac{1}{400} + \frac{1}{4,000} + \frac{1}{40,000} + \dots\\)
○ \\(\frac{4}{10} + \frac{4}{100} + \frac{4}{1,000} + \frac{4}{10,000} + \dots\\)
○ \\(\frac{1}{10} + \frac{1}{100} + \frac{1}{1,000} + \frac{1}{10,000} + \dots\\)

Explanation:

Step1: Analyze the decimal 0.4444...

The decimal \(0.4444\ldots\) can be written as the sum of its decimal places. The first digit after the decimal is \(4\) in the tenths place, the second is \(4\) in the hundredths place, the third is \(4\) in the thousandths place, and so on. So, \(0.4444\ldots=\frac{4}{10}+\frac{4}{100}+\frac{4}{1000}+\frac{4}{10000}+\ldots\)

Step2: Evaluate each option

• Option 1: \(\frac{1}{4}+\frac{1}{40}+\frac{1}{400}+\frac{1}{4000}+\ldots\) has a first term of \(\frac{1}{4} = 0.25\), not matching the decimal's first term \(0.4\).
• Option 2: \(\frac{1}{40}+\frac{1}{400}+\frac{1}{4000}+\frac{1}{40000}+\ldots\) has a first term of \(\frac{1}{40}=0.025\), not matching.
• Option 3: \(\frac{4}{10}+\frac{4}{100}+\frac{4}{1000}+\frac{4}{10000}+\ldots\) matches the expansion of \(0.4444\ldots\).
• Option 4: \(\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\frac{1}{10000}+\ldots\) has numerators of \(1\), not \(4\), so it would represent \(0.1111\ldots\), not \(0.4444\ldots\).

Answer:

\(\frac{4}{10}+\frac{4}{100}+\frac{4}{1,000}+\frac{4}{10,000}+\ldots\) (the third option)