Solution:
Let us take the LHS first:
We can write the given equation as:
91/3 + 1/9 + 1/27 + …∞
So let us take
$ m\text{ }=\text{ }1/3\text{ }+\text{ }1/9\text{ }+\text{ }1/27\text{ }+\text{ }\ldots \text{ }\infty $
Where on comparing we have
a = 1/3, r = (1/9) / (1/3) = 1/3
By making use of the formula,
$ {{S}_{\infty }}~=\text{ }a/\left( 1-r \right) $
$ =\text{ }\left( 1/3 \right)\text{ }/\text{ }\left( 1-\left( 1/3 \right) \right) $
$ =\text{ }\left( 1/3 \right)\text{ }/\text{ }\left( \left( 3-1 \right)/3 \right) $
$ =\text{ }\left( 1/3 \right)\text{ }/\text{ }\left( 2/3 \right) $
$ =1/2 $
So, 9m = 91/2 = 3 = RHS
Hence proved.