Solution:
We can write 0.125125125 as
0.125125125 = 0.125 + 0.000125 + 0.000000125 + …
This can be written as
$ 125/{{10}^{3}}~+\text{ }125/{{10}^{6}}~+\text{ }125/{{10}^{9}}~+\text{ }\ldots $
$125/{{10}^{3}}~\left[ 1\text{ }+\text{ }1/{{10}^{3}}~+\text{ }1/{{10}^{6}}~+\text{ }\ldots \right]$
By using the formula,
$ {{S}_{\infty }}~=\text{ }a/\left( 1-r \right) $
$ =125/{{10}^{3}}~\left[ 1\text{ }/\text{ }\left( 1-1/1000 \right) \right] $
$ =125/{{10}^{3}}~\left[ 1\text{ }/\text{ }\left( \left( 1000-1 \right)/1000 \right) \right)] $
$ =125/{{10}^{3}}[1\text{ }/\text{ }\left( 999/1000 \right)] $
$ =125/1000\text{ }\left( 1000/999 \right) $
$ =125/999 $
Therefore, we can represent the decimal 0.125125125 in form of a rational number as 125/999