## Patterns with 1/7

$1/7 = 0.142857142857142857142857142857142857142857142857142857142857...$

$0. \overline{142857}$   (period 6)

Move 7 :     $714285 \; = \; 5 \; \times \: 142857$

$2/7 = 0.285714285714285714285714285714285714285714285714285714285714...$

$0. \overline{285714}$   (period 6)

Move 4 :     $428571 \; = \; 3/2 \; \times \: 285714$

$3/7 = 0.428571428571428571428571428571428571428571428571428571428571...$

$0. \overline{428571}$   (period 6)

Move 1 :     $428571 \; = \; 3 \; \times \: 142857$

$4/7 = 0.571428571428571428571428571428571428571428571428571428571428...$

$0. \overline{571428}$   (period 6)

Move 8 :     $857142 \; = \; 3/2 \; \times \: 571428$

$5/7 = 0.714285714285714285714285714285714285714285714285714285714285...$

$0. \overline{714285}$   (period 6)

Move 5 :     $571428 \; = \; 4/5 \; \times \: 714285$

$6/7 = 0.857142857142857142857142857142857142857142857142857142857142...$

$0. \overline{857142}$   (period 6)

Move 2 :     $857142 \; = \; 3 \; \times \: 285714$

Other rotations:

$714285 \; = \; 5 \; \; \times \: \; 142857$
$571428 \; = \; 4 \; \; \times \: \; 142857$
$857142 \; = \; 6 \; \; \times \: \; 142857$
$285714 \; = \; 2 \; \; \times \: \; 142857$
$428571 \; = \; 3 \; \; \times \: \; 142857$

142857,   285714,   428571,   571428,   714285,   857142 :

$142857 \; \times \: \; 5 \; = \; 714285$
$714285 = 857^2 - 142^2$
where   $857142 = 142857 \; \times \: \; 6$

$714^2 \; - \; 285^2 \; = \; 428571$   where   $428571 = 3 \; \times \: \; 142857$

$571^2 \; - \; 428^2 \; = \; 142857$   where   $571428 = 4 \; \times \: \; 142857$

$142857^2 = 20408122449 = 20408||122449 = (20408 + 122449)^2$
$285714^2 = 81632489796 = 81632||489796$          $81632+489796 = 571428 = 4 \; \times \: 142857$
$428571^2 = 183673102041 = 183673||102041$      $183673+102041 = 285714 = 2 \; \times \: 142857$
$571428^2 = 326529959184 = 326529||959184$      $326529+959184 = 1285713 = 9 \; \times \: 142857$
$714285^2 = 510203061225 = 510203||061225$      $510203+061225 = 571428 = 4 \; \times \: 142857$
$857142^2 = 734692408164 = 734692||408164$      $734692+408164 = 1142856 = 8 \; \times \: 142857$

$7000007 \; \times \: 142857 \; \times \: 1 \; \; = \; 0999999999999$
$7000007 \; \times \: 142857 \; \times \: 2 \; \; = \; 1999999999998$
$7000007 \; \times \: 142857 \; \times \: 3 \; \; = \; 2999999999997$
$7000007 \; \times \: 142857 \; \times \: 4 \; \; = \; 3999999999996$
$7000007 \; \times \: 142857 \; \times \: 5 \; \; = \; 4999999999995$
$7000007 \; \times \: 142857 \; \times \: 6 \; \; = \; 5999999999994$
$7000007 \; \times \: 142857 \; \times \: 7 \; \; = \; 6999999999993$
$7000007 \; \times \: 142857 \; \times \: 8 \; \; = \; 7999999999992$
$7000007 \; \times \: 142857 \; \times \: 9 \; \; = \; 8999999999991$
$7000007 \; \times \: 142857 \; \times \: 10 \; \; = \; 9999999999990$

$7000007 \; \times \: 285714 \; \times \: 1 \; \; = \; \; 1999999999998$
$7000007 \; \times \: 285714 \; \times \: 2 \; \; = \; \; 3999999999996$
$7000007 \; \times \: 285714 \; \times \: 3 \; \; = \; \; 5999999999994$
$7000007 \; \times \: 285714 \; \times \: 4 \; \; = \; \; 7999999999992$
$7000007 \; \times \: 285714 \; \times \: 5 \; \; = \; \; 9999999999990$
$7000007 \; \times \: 285714 \; \times \: 6 \; \; = \; \; 11999999999988$
$7000007 \; \times \: 285714 \; \times \: 7 \; \; = \; \; 13999999999986$
$7000007 \; \times \: 285714 \; \times \: 8 \; \; = \; \; 15999999999984$
$7000007 \; \times \: 285714 \; \times \: 9 \; \; = \; \; 17999999999982$
$7000007 \; \times \: 285714 \; \times \: 10 \; \; = \; \; 19999999999980$

$7000007 \; \times \: \; 428571 \; \times \: \; 1 \; \; = \; \; 2999999999997$
$7000007 \; \times \: \; 428571 \; \times \: \; 2 \; \; = \; \; 5999999999994$
$7000007 \; \times \: \; 428571 \; \times \: \; 3 \; \; = \; \; 8999999999991$
$7000007 \; \times \: \; 428571 \; \times \: \; 4 \; \; = \; \; 11999999999988$
$7000007 \; \times \: \; 428571 \; \times \: \; 5 \; \; = \; \; 14999999999985$
$7000007 \; \times \: \; 428571 \; \times \: \; 6 \; \; = \; \; 17999999999982$
$7000007 \; \times \: \; 428571 \; \times \: \; 7 \; \; = \; \; 20999999999979$
$7000007 \; \times \: \; 428571 \; \times \: \; 8 \; \; = \; \; 23999999999976$
$7000007 \; \times \: \; 428571 \; \times \: \; 9 \; \; = \; \; 26999999999973$
$7000007 \; \times \: \; 428571 \; \times \: \; 10 \; \; = \; \; 29999999999970$

$7000007 \; \times \: \; 571428 \; \times \: \; 1 \; \; = \; \; 3999999999996$
$7000007 \; \times \: \; 571428 \; \times \: \; 2 \; \; = \; \; 7999999999992$
$7000007 \; \times \: \; 571428 \; \times \: \; 3 \; \; = \; \; 11999999999988$
$7000007 \; \times \: \; 571428 \; \times \: \; 4 \; \; = \; \; 15999999999984$
$7000007 \; \times \: \; 571428 \; \times \: \; 5 \; \; = \; \; 19999999999980$
$7000007 \; \times \: \; 571428 \; \times \: \; 6 \; \; = \; \; 23999999999976$
$7000007 \; \times \: \; 571428 \; \times \: \; 7 \; \; = \; \; 27999999999972$
$7000007 \; \times \: \; 571428 \; \times \: \; 8 \; \; = \; \; 31999999999968$
$7000007 \; \times \: \; 571428 \; \times \: \; 9 \; \; = \; \; 35999999999964$
$7000007 \; \times \: \; 571428 \; \times \: \; 10 \; \; = \; \; 39999999999960$

$7000007 \; \times \: \; 714285 \; \times \: \; 1 \; \; = \; \; 4999999999995$
$7000007 \; \times \: \; 714285 \; \times \: \; 2 \; \; = \; \; 9999999999990$
$7000007 \; \times \: \; 714285 \; \times \: \; 3 \; \; = \; \; 14999999999985$
$7000007 \; \times \: \; 714285 \; \times \: \; 4 \; \; = \; \; 19999999999980$
$7000007 \; \times \: \; 714285 \; \times \: \; 5 \; \; = \; \; 24999999999975$
$7000007 \; \times \: \; 714285 \; \times \: \; 6 \; \; = \; \; 29999999999970$
$7000007 \; \times \: \; 714285 \; \times \: \; 7 \; \; = \; \; 34999999999965$
$7000007 \; \times \: \; 714285 \; \times \: \; 8 \; \; = \; \; 39999999999960$
$7000007 \; \times \: \; 714285 \; \times \: \; 9 \; \; = \; \; 44999999999955$
$7000007 \; \times \: \; 714285 \; \times \: \; 10 \; \; = \; \; 49999999999950$

$7000007 \; \times \: \; 857142 \; \times \: \; 1 \; \; = \; \; 5999999999994$
$7000007 \; \times \: \; 857142 \; \times \: \; 2 \; \; = \; \; 11999999999988$
$7000007 \; \times \: \; 857142 \; \times \: \; 3 \; \; = \; \; 17999999999982$
$7000007 \; \times \: \; 857142 \; \times \: \; 4 \; \; = \; \; 23999999999976$
$7000007 \; \times \: \; 857142 \; \times \: \; 5 \; \; = \; \; 29999999999970$
$7000007 \; \times \: \; 857142 \; \times \: \; 6 \; \; = \; \; 35999999999964$
$7000007 \; \times \: \; 857142 \; \times \: \; 7 \; \; = \; \; 41999999999958$
$7000007 \; \times \: \; 857142 \; \times \: \; 8 \; \; = \; \; 47999999999952$
$7000007 \; \times \: \; 857142 \; \times \: \; 9 \; \; = \; \; 53999999999946$
$7000007 \; \times \: \; 857142 \; \times \: \; 10 \; \; = \; \; 59999999999940$

$1 \; \times \: \; 7 \; + \; 3 \; = \; 10$
$14 \; \times \: \; 7 \; + \; 2 \; = \; 100$
$142 \; \times \: \; 7 \; + \; 6 \; = \; 1000$
$1428 \; \times \: \; 7 \; + \; 4 \; = \; 10000$
$14285 \; \times \: \; 7 \; + \; 5 \; = \; 100000$
$142857 \; \times \: \; 7 \; + \; 1 \; = \; 1000000$
$1428571 \; \times \: \; 7 \; + \; 3 \; = \; 10000000$
$14285714 \; \times \: \; 7 \; + \; 2 \; = \; 100000000$
$142857142 \; \times \: \; 7 \; + \; 6 \; = \; 1000000000$
$1428571428 \; \times \: \; 7 \; + \; 4 \; = \; 10000000000$
$14285714285 \; \times \: \; 7 \; + \; 5 \; = \; 100000000000$
$142857142857 \; \times \: \; 7 \; + \; 1 \; = \; 1000000000000$
$1428571428571 \; \times \: \; 7 \; + \; 3 \; = \; 10000000000000$
$14285714285714 \; \times \: \; 7 \; + \; 2 \; = \; 100000000000000$
$142857142857142 \; \times \: \; 7 \; + \; 6 \; = \; 1000000000000000$
$1428571428571428 \; \times \: \; 7 \; + \; 4 \; = \; 10000000000000000$
$14285714285714285 \; \times \: \; 7 \; + \; 5 \; = \; 100000000000000000$
$142857142857142857 \; \times \: \; 7 \; + \; 1 \; = \; 1000000000000000000$

$2 \; \times \: \; 7 \; + \; 6 \; = \; 20$
$28 \; \times \: \; 7 \; + \; 4 \; = \; 200$
$285 \; \times \: \; 7 \; + \; 5 \; = \; 2000$
$2857 \; \times \: \; 7 \; + \; 1 \; = \; 20000$
$28571 \; \times \: \; 7 \; + \; 3 \; = \; 200000$
$285714 \; \times \: \; 7 \; + \; 2 \; = \; 2000000$
$2857142 \; \times \: \; 7 \; + \; 6 \; = \; 20000000$
$28571428 \; \times \: \; 7 \; + \; 4 \; = \; 200000000$
$285714285 \; \times \: \; 7 \; + \; 5 \; = \; 2000000000$
$2857142857 \; \times \: \; 7 \; + \; 1 \; = \; 20000000000$
$28571428571 \; \times \: \; 7 \; + \; 3 \; = \; 200000000000$
$285714285714 \; \times \: \; 7 \; + \; 2 \; = \; 2000000000000$
……………………..

$4 \; \times \: \; 7 \; + \; 2 \; = \; 30$
$42 \; \times \: \; 7 \; + \; 6 \; = \; 300$
$428 \; \times \: \; 7 \; + \; 4 \; = \; 3000$
$4285 \; \times \: \; 7 \; + \; 5 \; = \; 30000$
$42857 \; \times \: \; 7 \; + \; 1 \; = \; 300000$
$428571 \; \times \: \; 7 \; + \; 3 \; = \; 3000000$
………………………..

$5 \; \times \: \; 7 \; + \; 5 \; = \; 40$
$57 \; \times \: \; 7 \; + \; 1 \; = \; 400$
$571 \; \times \: \; 7 \; + \; 3 \; = \; 4000$
$5714 \; \times \: \; 7 \; + \; 2 \; = \; 40000$
$57142 \; \times \: \; 7 \; + \; 6 \; = \; 400000$
$571428 \; \times \: \; 7 \; + \; 4 \; = \; 4000000$
……………………

$7 \; \times \: \; 7 \; + \; 1 \; = \; 50$
$71 \; \times \: \; 7 \; + \; 3 \; = \; 500$
$714 \; \times \: \; 7 \; + \; 2 \; = \; 5000$
$7142 \; \times \: \; 7 \; + \; 6 \; = \; 50000$
$71428 \; \times \: \; 7 \; + \; 4 \; = \; 500000$
$714285 \; \times \: \; 7 \; + \; 5 \; = \; 5000000$
……………………

$8 \; \times \: \; 7 \; + \; 4 \; = \; 60$
$85 \; \times \: \; 7 \; + \; 5 \; = \; 600$
$857 \; \times \: \; 7 \; + \; 1 \; = \; 6000$
$8571 \; \times \: \; 7 \; + \; 3 \; = \; 60000$
$85714 \; \times \: \; 7 \; + \; 2 \; = \; 600000$
$857142 \; \times \: \; 7 \; + \; 6 \; = \; 6000000$
…………………….