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UPSC CDS 1 2022 Math Paper

Question : 40 of 100

Marks: +1, -0

\;\frac{1}{5\sqrt{4}+4\sqrt{5}}+\;\frac{1}{6\sqrt{5}+5\sqrt{6}}+\;\frac{1}{7\sqrt{6}+6\sqrt{7}}+\;\frac{1}{8\sqrt{7}+7\sqrt{8}}+\;\frac{1}{9\sqrt{8}+8\sqrt{9}}

Solution: 👈: Video Solution

Given:

\;\frac{1}{5\sqrt{4}+4\sqrt{5}}+\;\frac{1}{6\sqrt{5}+5\sqrt{6}}+\;\frac{1}{7\sqrt{6}+6\sqrt{7}}+\;\frac{1}{8\sqrt{7}+7\sqrt{8}}+\;\frac{1}{9\sqrt{8}+8\sqrt{9}}

Concept used:

(a+b)(a-b)=a^{2}-b^{2}

Calculation:

Consider the first term only, i.e.

\;\frac{1}{5\sqrt{4}+4\sqrt{5}}

Rationalize the denominator

\Rightarrow \;\frac{1}{5\sqrt{4}+4\sqrt{5}} \times \frac{5\sqrt{4}-4\sqrt{5}}{5\sqrt{4}-4\sqrt{5}}

\Rightarrow \;\frac{5\sqrt{4}-4\sqrt{5}}{(5\sqrt{4})^{2}-(4\sqrt{5})^{2}}

\Rightarrow \;\frac{5\sqrt{4}}{20}-\;\frac{4\sqrt{5}}{20}

\Rightarrow \;\frac{1}{\sqrt{4}}-\;\frac{1}{\sqrt{5}}

After doing a similar operation in other terms, we will get

\;\frac{1}{\sqrt{4}}-\;\frac{1}{\sqrt{5}}+\;\frac{1}{\sqrt{5}}-\;\frac{1}{\sqrt{6}}+\;\frac{1}{\sqrt{6}}-\;\frac{1}{\sqrt{7}}+\;\frac{1}{\sqrt{7}}-\;\frac{1}{\sqrt{8}}+\;\frac{1}{\sqrt{8}}-\;\frac{1}{\sqrt{9}}

\Rightarrow \;\frac{1}{\sqrt{4}}-\;\frac{1}{\sqrt{9}}

\Rightarrow \;\frac{1}{2}-\;\frac{1}{3}

\Rightarrow \;\frac{3-2}{6}

\Rightarrow \;\frac{1}{6}

\therefore

Required value of the expression is

\;\frac{1}{6}

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