Reciprocal of 89

From ProofWiki
Jump to navigation Jump to search

Theorem

The decimal expansion of the reciprocal of $89$ contains the Fibonacci sequence:

$\dfrac 1 {89} = 0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$

This sequence is A021093 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division:

   0.0112359550561797752808988764044943820224719101...
  ------------------------------------------------------
89)1.0000000000000000000000000000000000000000000000000
     89       534       712       801        89
     --       ---       ---       ---       ---  
     110       160       880       390       810
      89        89       801       356       801
     ---       ---       ---       ---       ---
      210       710       790       340        90
      178       623       712       267        89
      ---       ---       ---       ---        --
       320       870       780       730        100
       267       801       712       712         89
       ---       ---       ---       ---        --- 
        530       690       680       180       ...
        445       623       623       178
        ---       ---       ---       ---
         850       670       570        200
         801       623       534        178
         ---       ---       ---        ---
          490       470       360        220
          445       445       356        178
           --       ---       ---        ---
           450       250        400       420
           445       178        356       356
           ---       ---        ---       ---
             500      720        440       640
             445      712        356       623
             ---      ---        ---       ---
              550       800       840       170
              534       712       801        89

$\blacksquare$


Sources