To express the repeating decimal \( 0.414141... \) as a fraction, we can use the following method.
Let \( x = 0.414141... \).
Since the repeating part has two digits ("41"), we can multiply \( x \) by \( 100 \) to shift the decimal point two places to the right:
\[ 100x = 41.414141... \]
Now, we subtract the original equation from this new equation:
\[ 100x - x = 41.414141... - 0.414141... \] \[ 99x = 41 \]
Next, we solve for \( x \):
\[ x = \frac{41}{99} \]
Thus, the repeating decimal \( 0.414141... \) can be expressed as the fraction \( \frac{41}{99} \).
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