Reciprocal of 49

Theorem

The decimal expansion of the reciprocal of $49$ is as follows:

$\dfrac 1 {49} = 0 \cdotp \dot 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 5 \dot 1$

This sequence is A021053 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

Performing the calculation using long division:

   0.02040816326530612244897959183673469387755102...
  ---------------------------------------------
49)1.00000000000000000000000000000000000000000000000
     98      294     196    441    196    245 
     --      ---     ---    ---    ---    ---
      200     260     240     90    340    250
      196     245     196     49    294    245
      ---     ---     ---     --    ---    ---
        400    150     440    410    460     50
        392    147     392    392    441     49
        ---    ---     ---    ---    ---     --
          80     300    480    180    190     100
          49     294    441    147    147      98
          --     ---    ---    ---    ---     ---
          310      60    390    330    430    ...
          294      49    343    294    392
          ---      --    ---    ---    ---
           160     110    470    360    380
           147      98    441    343    343
           ---     ---    ---    ---    ---
            130     120    290    170    360
             98      98    245    147    343
             ---    ---    ---    ---    ---
             320     220    450    230    270
             294     196    441    196    245
             ---     ---    ---    ---    ---

$\blacksquare$