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Question
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
Solution
\[\text { LHS }= 9^\frac{1}{3} . 9^\frac{1}{9} . 9^\frac{1}{27} . . . \infty \]
\[ = 9^\left( \frac{1}{3} + \frac{1}{9}\frac{1}{27} . \right) \]
\[ = 9^\left\{ \frac{\left( \frac{1}{3} \right)}{\left( 1 - \frac{1}{3} \right)} \right\} \]
\[ = 9^\frac{\left( \frac{1}{3} \right)}{\left( 1 - \frac{1}{3} \right)} \]
\[ = \sqrt{9}\]
\[ = 3 =\text { RHS }\]
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