The value of `(1)/(1+sqrt(2))+(1)/(sqrt(2)+sqrt(3))+(1)/(sqrt(3)+sqrt(4))+....+(1)/(sqrt(8)+sqrt(9))` is

Options

A. `1`
B. `0`
C. `2`
D. `sqrt(2)`

Answer

To solve the expression

\[ S = \frac{1}{1+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{4}} + \ldots + \frac{1}{\sqrt{8}+\sqrt{9}}, \]

we can simplify each term in the series. The general term can be expressed as:

\[ \frac{1}{\sqrt{n} + \sqrt{n+1}}. \]

To simplify this, we can rationalize the denominator. We multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{1}{\sqrt{n} + \sqrt{n+1}} \cdot \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+1} - \sqrt{n}} = \frac{\sqrt{n+1} - \sqrt{n}}{(\sqrt{n+1})^2 - (\sqrt{n})^2}. \]

The denominator simplifies to:

\[ (\sqrt{n+1})^2 - (\sqrt{n})^2 = (n+1) - n = 1. \]

Thus, we have:

\[ \frac{1}{\sqrt{n} + \sqrt{n+1}} = \sqrt{n+1} - \sqrt{n}. \]

Now, substituting this back into our series, we get:

\[ S = (\sqrt{2} - 1) + (\sqrt{3} - \sqrt{2}) + (\sqrt{4} - \sqrt{3}) + (\sqrt{5} - \sqrt{4}) + (\sqrt{6} - \sqrt{5}) + (\sqrt{7} - \sqrt{6}) + (\sqrt{8} - \sqrt{7}) + (\sqrt{9} - \sqrt{8}). \]

Notice that this is a telescoping series. Most terms will cancel out:

\[ S = -1 + \sqrt{9} = -1 + 3 = 2. \]

Thus, the value of the series is:

\[ \boxed{2}. \]