Question

$\textsf{The value of expression }\\\\ \displaystyle \sqrt{\dfrac{1}{\sqrt{2} + \sqrt{1}}+\dfrac{1}{\sqrt{3}+\sqrt{2}}+\dfrac{1}{\sqrt{4}+\sqrt{3}}+...........\sf upto \ 99 \ terms} \\\\\textsf{is equal to:}\\\\A)9\\B)3\\C)1\\D)0\\\\\textsf{Explain step by step!}$

Answer 1

Answer:

→B = 3.

Step-by-step explanation:

We have,

$\because \sf \sqrt{\dfrac{1}{\sqrt{2} + \sqrt{1}}+\dfrac{1}{\sqrt{3}+\sqrt{2}}+\dfrac{1}{\sqrt{4}+\sqrt{3}}+...........\sf upto \ 99 \ terms} \\ \\ \sf = \sqrt{ \frac{1}{ \sqrt{2} + \sqrt{1} } \times \frac{ \sqrt{2} - \sqrt{1} }{ \sqrt{2} - \sqrt{1} } + \frac{1}{ \sqrt{3} + \sqrt{2} } \times \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} - \sqrt{2} } + ........ + \frac{1}{ \sqrt{100} + \sqrt{99} } \times \frac{ \sqrt{100} - \sqrt{99} }{ \sqrt{100} - \sqrt{99} } } . \\ \\ \sf = \sqrt{ \frac{ \sqrt{2} - 1 }{2 - 1} + \frac{ \sqrt{3} - \sqrt{2} }{3 - 2} + ...... + \frac{10 - \sqrt{99} }{100 - 99} } . \\ \\ \sf = \sqrt{ \cancel{\sqrt{2}} - 1 + \cancel{\sqrt{3}} - \cancel{\sqrt{2}} + \cancel{\sqrt{4}} - \cancel{\sqrt{3}} + .....10 - \cancel{\sqrt{99} } } . \\ \\ = \sf \sqrt{ - 1 + 10} . \\ \\ \sf = \sqrt{9} . \\ \\ \huge \pink{ \boxed{ \boxed{ \it = 3.}}}$

Correct option :- B) 3 .

Hence, it is solved.

Answer 2

$\mathfrak{\large{\underline{\underline{Answer:-}}}}$

Option => B

The value of that expression is 3.

$\mathfrak{\large{\underline{\underline{Explanation:-}}}}$

Solution :

Given expression :

$\sf \sqrt{\dfrac{1}{\sqrt{2} + \sqrt{1}}+\dfrac{1}{\sqrt{3}+\sqrt{2}}+\dfrac{1}{\sqrt{4}+\sqrt{3}}+...........\sf upto\;99\;terms...}$

So ,

$\sf \sqrt{\dfrac{1}{\sqrt{2}+\sqrt{1}}\;\times\dfrac{\sqrt{2} -\sqrt{1}}{\sqrt{2} -\sqrt{1}}\;+\dfrac{1}{\sqrt{3}+\sqrt{2}}\;\times\dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}+.......+\dfrac{\sqrt{1}}{\sqrt{100} +\sqrt{99}}\times\dfrac{\sqrt{100}-\sqrt{99}}{\sqrt{100}-\sqrt{99}}}$

$\sf= \sqrt{ \frac{ \sqrt{2} - 1 }{2 - 1} + \frac{ \sqrt{3} -\sqrt{2} }{3 - 2} + ......+\dfrac{10 - \sqrt{99} }{100 - 99}}$

Hence ,

$\sf = \sqrt{\cancel{\sqrt{2}} -1 + \cancel{\sqrt{3}} - \cancel{\sqrt{2}} +\cancel{\sqrt{4}}- \cancel{\sqrt{3}}+..10 \cancel{\sqrt{99}}}$

$\sf \sqrt{-1 + 10}$

$\sf =\sqrt{9}=3$

Hence,

Correct Option => B = 3.