SOLUTION: Express the nonterminating repeating decimal 2.045045045... as a fraction.

N = 2.045045045...

On the right, move the decimal just to the right of the end of the first cycle
of repeating digits, multiply the left side by the power of 10 corresponding to
the moving of the decimal on the right.

In other words, multiply both sides by 1000

1000N = 2045.045045045...

Subtract the original equation

1000N = 2045.045045045...
    N =    2.045045945...
-------------------------
 999N = 2043
    N = 2043/999
    N = 227/111

Edwin


 

Since you posted this in the Sequences-and-series section, maybe you wanted to
see it done as an infinite geometric series:

2 + .045
  + .000045
  + .000000045
  + ...........

2 + 0.045 + 0.000045 + 0.000000045 + ...

2 + 4.5x10-2 + 4.5x10-5 + 4.5x10-8 + ...

The terms to the right of 2 form an infinite geometric series with first term =
a1 = 4.5x10-2 and common ratio r = 10-3





Multiply numerator and denominator by 103





So the fraction is  or 

Edwin