Let us define x=0.142857142857... and consider 10^12*x to get
10^12x=142857142857.142857142857...
Now, we subtract the original one from the one we have multiplied to get
10^12*x - x = 142857142857.00000
So, using distribution rule
(10^12 - 1)*x = 142857142857.00000
99999999999*x = 142857142857
We can use our calculator to see that
99999999999/142857142857 = 7
Thus,
x = 1/7.