Question

Find the sum up to 91 terms of the following
$\frac{1}{\sqrt{9}+\sqrt{10}}+\frac{1}{\sqrt{10}+\sqrt{11}}+\frac{1}{\sqrt{11}+\sqrt{12}}+.....$

• A. $90$
• B. $10$
• C. $7$
• D. $1$

Answer 1

The correct option is C $7$

$\frac{1}{\sqrt{9}+\sqrt{10}}+\frac{1}{\sqrt{10}+\sqrt{11}}+\frac{1}{\sqrt{11}+\sqrt{12}}+....+\frac{1}{\sqrt{90}+\sqrt{100}}.$

Rationalising...we get

$\frac{\sqrt{10}-\sqrt{9}}{(\sqrt{9}+\sqrt{10})(\sqrt{11}-\sqrt{10})}+\frac{\sqrt{11}-\sqrt{10}}{(\sqrt{10}+\sqrt{11})(\sqrt{11}-\sqrt{10})}+....+\frac{\sqrt{100}-\sqrt{99}}{(\sqrt{90}+\sqrt{100})(\sqrt{100}-\sqrt{99}}.$

=$\sqrt{10}-\sqrt{9}+\sqrt{11}-\sqrt{10}+\sqrt{12}-\sqrt{11}+.............+\sqrt{100}-\sqrt{99}$

=$\sqrt{100}-\sqrt{9}$

=$10-3$

=$7$