Show that $0.142857142857... = 0.\overline{142857} = \frac{1}{7}$.

(N/A) Let $x = 0.142857142857...$ (Equation $1$).
Since there are $6$ repeating digits,multiply both sides of Equation $1$ by $10^6 = 1,000,000$:
$1,000,000x = 142857.142857142857...$ (Equation $2$).
Subtract Equation $1$ from Equation $2$:
$1,000,000x - x = 142857.142857... - 0.142857...$
$999,999x = 142857$.
$x = \frac{142857}{999999}$.
Dividing both numerator and denominator by $142857$,we get:
$x = \frac{1}{7}$.
Thus,$0.\overline{142857} = \frac{1}{7}$.