Wow! This pattern that you have displayed actually continues past 7, actually, as I discovered just now. You just have to perform an intermediary manipulation.
142857∗8=1142856⇒1+142856=142857
142857∗9=1285713⇒1+285713=285714
142857∗10=1428570⇒1+428570=428571
It seems to be that any multiple of 7 results in the 999999 result while anything else results in this "cyclic" property.
142857∗77=10999989⇒10+999989=999999
Here are a few arbitrary attempts I tried to see if the pattern maintained itself. To my relief, it does.
142857∗43=6142851⇒6+142851=142857
142857∗6807=972427599⇒972+427599=428571
142857∗142857=20408122449⇒20408+122449=142857
Amazingly, this pattern maintains itself for incredibly large multipliers. It requires more manipulation, though. This is incredible!
142857∗758241142857=108320054945122449⇒108320054945+122449=108320177394108320177394⇒108320+177394=285714
.