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Question
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
Solution
\[\text { LHS } = 2^\frac{1}{4} . 4^\frac{2}{8} . 8^\frac{3}{16} . {16}^\frac{4}{32} . . . \infty \]
\[ = 2^\left( \frac{1}{4} + \frac{2}{8} + \frac{3}{16}\frac{3}{16}\frac{4}{32} . \infty \right) \]
\[ = 2^\left( \frac{1}{2^2} + \frac{2}{2^3} + \frac{3}{2^4} + \frac{4}{2^5} + . . . \infty \right) \]
\[ = 2^\frac{1}{2^2}\left\{ 1 + \frac{2}{2} + \frac{3}{2^2} + \frac{4}{2^3} . . . \infty \right\} \]
\[ = 2^\frac{1}{2^2}\left\{ \frac{1}{1 - \frac{1}{2}} + \frac{1 . \frac{1}{2}}{\left( 1 - \frac{1}{2} \right)^2} \right\} \]
\[ = 2^\frac{1}{2^2}\left\{ 2 + 2 \right\} \]
\[ = 2^1 \]
\[ = 2 = \text { RHS }\]
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