Question

The infinite geometric sum formula can be used to write 0.126126126 . . . as a fraction. What is the numerator of this reduced fraction?
(1 point)
Responses

7
7

14
14

42
42

126

Answer 1

To express the repeating decimal \(0.126126126\ldots\) as a fraction, we can use the formula for the sum of an infinite geometric series.

First, let's identify the repeating part:
- The repeating part is \(126\), which repeats every 3 digits.

We can express \(0.126126126\ldots\) as:

\[
0.126126126\ldots = 0.126 + 0.000126 + 0.000000126 + \cdots
\]

This series can be viewed as a geometric series where:
- The first term \(a = 0.126\)
- The common ratio \(r = 0.001\) (because each term is \(1/1000\) of the previous term)

The sum \(S\) of an infinite geometric series can be calculated using the formula:

\[
S = \frac{a}{1 - r}
\]

Plugging in the values:

\[
S = \frac{0.126}{1 - 0.001} = \frac{0.126}{0.999}
\]

Next, we can convert \(0.126\) and \(0.999\) into fractions:

\[
0.126 = \frac{126}{1000}
\]
\[
0.999 = \frac{999}{1000}
\]

Therefore, substituting these values into the sum formula gives us:

\[
S = \frac{\frac{126}{1000}}{\frac{999}{1000}} = \frac{126}{999}
\]

Now, we need to simplify this fraction. The greatest common divisor (GCD) of \(126\) and \(999\) can be found. Both numbers can be divided by \(9\):

\[
126 \div 9 = 14
\]
\[
999 \div 9 = 111
\]

Thus, we can simplify the fraction:

\[
\frac{126}{999} = \frac{14}{111}
\]

This fraction is in reduced form where \(14\) is the numerator.

Hence, the numerator of the reduced fraction is:

\[
\boxed{14}
\]