The value of $$\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+......+\frac{1}{\sqrt{8}+\sqrt{9}}$$ is

Expression : $$\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+......+\frac{1}{\sqrt{8}+\sqrt{9}}$$

After rationalizing, the denominator of each term will be 1, the numerator will be

= $$\sqrt{2}$$ - 1 + $$\sqrt{3}$$ - $$\sqrt{2}$$ + $$\sqrt{4}$$ - $$\sqrt{3}$$ +.......+ $$\sqrt{8}$$ - $$\sqrt{7}$$ + $$\sqrt{9}$$ - $$\sqrt{8}$$

Now, all the terms will cancel out except

= $$\sqrt{9}$$ - 1 = 3-1

= 2