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UPSC CDS 2 2020 Math Paper

Question : 69 of 100

Marks: +1, -0

\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}

+ \dots + \frac{1}{\sqrt{99}+\sqrt{100}} = ?

Solution:

we have

\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}

+ \dots + \frac{1}{\sqrt{99}+\sqrt{100}}

Rationalization of each term

\Rightarrow \frac{\sqrt{2}-1}{2-1} + \frac{\sqrt{3}-\sqrt{2}}{3-2} + \frac{\sqrt{4}-\sqrt{3}}{4-3}

+ \dots + \frac{\sqrt{100}-\sqrt{99}}{100-99}

\Rightarrow(\sqrt{2}-1)+(\sqrt{3}-\sqrt{2})+(\sqrt{4}-\sqrt{3})

+ \dots + (\sqrt{100}-\sqrt{99})

\Rightarrow\sqrt{100}-1

\Rightarrow10-1

\Rightarrow9

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