NIMCET 2016 — Mathematics PYQ

Calculation: 

$\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\cdots +\frac{1}{\sqrt{80}+\sqrt{81}}$

Rationalize the numerator and denominator  

$=\frac{1}{\sqrt{1}+\sqrt{2}}\times \frac{\sqrt{2}-\sqrt{1}}{\sqrt{2}-\sqrt{1}}+\frac{1}{\sqrt{2}+\sqrt{3}}\times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}\times \frac{\sqrt{4}-\sqrt{3}}{\sqrt{4}-\sqrt{3}}+\cdots +\frac{1}{\sqrt{80}+\sqrt{81}}\times \frac{\sqrt{81}-\sqrt{80}}{\sqrt{81}-\sqrt{80}}$

$=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+\cdots +\sqrt{80}-\sqrt{79}+\sqrt{81}-\sqrt{80}$  

$=\sqrt{81}-\sqrt{1}$  

$=9-1$  

$=8$  

$\therefore \frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\cdots +\frac{1}{\sqrt{80}+\sqrt{81}}=8$