Let us consider the LHS
(91/3 . 91/9 . 91/27 ….∞)
This can be written as
91/3 + 1/9 + 1/27 + …∞
So let us consider m = 1/3 + 1/9 + 1/27 + … ∞
Where, a = 1/3, r = (1/9) / (1/3) = 1/3
By using the formula,
S∞ = a/(1 – r)
= (1/3) / (1 – (1/3))
= (1/3) / ((3 - 1)/3)
= (1/3) / (2/3)
= 1/2
So, 9m = 91/2 = 3 = RHS
Hence proved.