Question

Prove that :
(9^1/3 . 9^1/9 . 9^1/27 ….∞) = 3.

Answer 1

Let us consider the LHS

(9^1/3 . 9^1/9 . 9^1/27 ….∞)

This can be written as

9^{1/3 + 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, 9^m = 9^1/2 = 3 = RHS

Hence proved.