Reciprocal of 29

Theorem

The decimal expansion of the reciprocal of $29$ is as follows:

$\dfrac 1 {29} = 0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$

This sequence is A021033 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division: 

   0.034482758620689655172413793103...
  ------------------------------------
29)1.000000000000000000000000000000...
     87
     -- 
     130    180     150    230
     116    174     145    203
     ---    ---     ---    ---
      140     60      50    270
      116     58      29    261
      ---     --      --    ---
       240     200    210     90
       232     174    203     87
       ---     ---    ---     --
         80     260     70     30
         58     232     58     29
         --     ---     --     --
         220     280    120     100
         203     261    116      87
          --     ---    ---     ---
          170     190     40    ...
          145     174     29
          ---     ---     --
           250     160    110
           232     145     87
           ---     ---    ---
            180     150    230

$\blacksquare$

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