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:
\[ \boxed{14} \]
1 answer