Question

Prove that 2^1/4.4^1/8.8^1/16.16^1/32.............is equal to 2

Answer 1

$\text{P} = 2^\frac{1}{4} \text{x} 4^\frac{1}{8} \text{x}8^\frac{1}{16}\text{x}16^\frac{1}{32}...............\text{infinite} = 2$ is Proved.

Step-by-step explanation:

To Prove :

$\text{P} = 2^\frac{1}{4} \text{x} 4^\frac{1}{8} \text{x}8^\frac{1}{16}\text{x}16^\frac{1}{32}...............\text{infinite} = 2$

P = $2^({\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{64}.....) }$

P =$2^\text{s}$ ...........(1)

where S = ${\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{5}{64}..... }$ (2)

$\text{S}\text{x}\frac{1}{2} = \frac{1}{8} +\frac{2}{16} + \frac{3}{32} +\frac{4}{64} ................$(3)

subtracting equation (3) from (2)

$\frac{\text{S}}{2} =\frac{1}{4} +\frac{1}{8} +\frac{1}{32} +\frac{1}{64} .........$

$\frac{\text{S}}{2} = \frac{\text{a}}{(1-\text{r})}$

   = $\frac{1}{4} \text{x}\frac{2}{1} = \frac{1}{2}$

∴ S = 1

Substitute S = 1 in equation (1)

$\text{P} = 2^{1} = 2$

Hence proved.

To Learn More....

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