214. 418. 8116. 16132................ is equal to

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2^{\frac{1}{4}}

.

4^{\frac{1}{8}}

.

8^{\frac{1}{16}}

.

16^{\frac{1}{32}}

................ is equal to

Options:

A . 1

B . 2

C . 32

D . 52

Answer: Option B
:
B

2^{\frac{1}{4}}

.

4^{\frac{1}{8}}

.

8^{\frac{1}{16}}

.

16^{\frac{1}{32}}

.........∞
=

2^{\frac{1}{4} + \frac{2}{8} + \frac{3}{16} + . . . . . . . .}

=

2^{S}

, where S is given by
S =

\frac{1}{4}

+ 2

\frac{1}{8}

+ 3

\frac{1}{16}

+4

\frac{1}{32}

+ ............∞ .........(i)
⇒

\frac{1}{2}

S =

\frac{1}{8}

+

\frac{2}{16}

+

\frac{3}{32}

+

\frac{4}{64}

+ ................∞ ..........(ii)
Subtracting (ii) from (i), we get S = 1.
Hence required product =

2^{1}

= 2.

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