Prove that:

Question:

Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.

Solution:

$\mathrm{LHS}=9^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 9^{\frac{1}{27} \ldots \infty}$

$=9\left(\frac{1}{3}+\frac{1}{9} \frac{1}{27} .\right)$

$=9\left\{\frac{\left(\frac{1}{3}\right)}{\left(1-\frac{1}{3}\right)}\right\}$

$=9^{\frac{\left(\frac{1}{3}\right)}{\left(1-\frac{1}{3}\right)}}$

$=\sqrt{9}$

$=3=\mathrm{RHS}$

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