Simplify : 1 3 + 4 \,+ 1 4 + 5 \,+ 1 5 + 6 \,+ 1 6 + 7 \,+ 1 7 + 8 \, + 1 8 + 9 = ?

Answer: Option C

$$\begin{array}{rl}& \frac{1}{\sqrt{3}+\sqrt{4}}\\ & =\frac{1}{\sqrt{4}+\sqrt{3}}\times \frac{\sqrt{4}-\sqrt{3}}{\sqrt{4}-\sqrt{3}}\\ & =\frac{\sqrt{4}-\sqrt{3}}{1}\\ & =\sqrt{4}-\sqrt{3}\\ & \text{Similarly}\\ & \frac{1}{\sqrt{4}+\sqrt{5}}=\sqrt{5}-\sqrt{4}\\ & \Rightarrow \frac{1}{\sqrt{5}+\sqrt{6}}=\sqrt{6}-\sqrt{5}\\ & \Rightarrow \frac{1}{\sqrt{6}+\sqrt{7}}=\sqrt{7}-\sqrt{6}\\ & \Rightarrow \frac{1}{\sqrt{7}+\sqrt{8}}=\sqrt{8}-\sqrt{7}\\ & \Rightarrow \frac{1}{\sqrt{8}+\sqrt{9}}=\sqrt{9}-\sqrt{8}\\ & \text{Now put values}\\ & \Rightarrow \sqrt{4}-\sqrt{3}+\sqrt{5}-\sqrt{4}+\sqrt{6}-\sqrt{5}+\sqrt{7}-\sqrt{6}+\sqrt{8}-\sqrt{7}+\sqrt{9}-\sqrt{8}\\ & \Rightarrow \sqrt{9}-\sqrt{3}\\ & \Rightarrow 3-\sqrt{3}\end{array}$$