First slide

Arithmetico-Geometric Progression

Question

$2^{1 / 4} \cdot 4^{1 / 8} \cdot 8^{1 / 16} \dots \infty is equal to$

Difficult

A

1
B

2
C

$\frac{3}{4}$
D

4

Solution

$S = \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \dots \infty \to (1)$
$\begin{array}{l} \frac{1}{2} S = \frac{1}{8} + \frac{2}{16} + \frac{3}{32} + \dots \infty - \dots \to (2) \\ (1) - (2) \Rightarrow \\ S - \frac{1}{2} S = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + . . . . . . \\ = \frac{\frac{1}{4}}{1 - 1 / 2} = 1 / 2 (s u m o f i n f i n i t e t e r m s i n G . P = \frac{a}{1 - r}) \\ \frac{1}{2} S = \frac{1}{2} \to S = 1 \\ g i v e n s u m = 2^{S} = 2^{1} = 2 . \end{array}$

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