Reciprocal of 71

Theorem

The decimal expansion of the reciprocal of $71$ is as follows:

$\dfrac 1 {71} = 0 \cdotp \dot 01408 \, 45070 \, 42253 \, 52112 \, 67605 \, 63380 \, 2816 \dot 9$

This sequence is A021075 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

Performing the calculation using long division:

   0.0140845070422535211267605633802816901...
  -------------------------------------------
71)1.0000000000000000000000000000000000000...
     71
   ----
     290      160    80    400    580
     284      142    71    355    568
     ---      ---    --    ---    ---
       600     180    90    450    120
       568     142    71    426     71
       ---     ---    --    ---    ---
        320     380   190    240    490
        284     355   142    213    426
        ---     ---   ---    ---    ---
         360     250   480    270    640
         355     213   426    213    639
         ---     ---   ---    ---    ---
           500    370   540    570     100
           497    355   497    568      71
           ---    ---   ---    ---     ---
             300   150   430     200   ...
             284   142   426     142
             ---   ---   ---     ---
              160    80    400    580

$\blacksquare$