The value of `(1)/(sqrt(2)+1)+(1)/(sqrt(3)+sqrt(2))+(1)/(sqrt(4)+sqrt(3))+...+(1)/(sqrt(100)+sqrt(99))` is

Options

A. `1`
B. `9`
C. `sqrt(99)`
D. `sqrt(99)-1`

Answer

To solve the expression

\[ S = \frac{1}{\sqrt{2}+1} + \frac{1}{\sqrt{3}+\sqrt{2}} + \frac{1}{\sqrt{4}+\sqrt{3}} + \ldots + \frac{1}{\sqrt{100}+\sqrt{99}}, \]

we will rationalize each term in the sum.

### Step 1: Rationalize the first term

The first term is

\[ \frac{1}{\sqrt{2}+1}. \]

To rationalize it, we multiply the numerator and the denominator by \(\sqrt{2}-1\):

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

### Step 2: Rationalize the second term

The second term is

\[ \frac{1}{\sqrt{3}+\sqrt{2}}. \]

Rationalizing it by multiplying by \(\sqrt{3}-\sqrt{2}\):

\[ \frac{1}{\sqrt{3}+\sqrt{2}} \cdot \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}} = \frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3})^2 - (\sqrt{2})^2} = \frac{\sqrt{3}-\sqrt{2}}{3-2} = \sqrt{3}-\sqrt{2}. \]

### Step 3: Rationalize the third term

The third term is

\[ \frac{1}{\sqrt{4}+\sqrt{3}}. \]

Rationalizing it:

\[ \frac{1}{\sqrt{4}+\sqrt{3}} \cdot \frac{\sqrt{4}-\sqrt{3}}{\sqrt{4}-\sqrt{3}} = \frac{\sqrt{4}-\sqrt{3}}{(\sqrt{4})^2 - (\sqrt{3})^2} = \frac{2-\sqrt{3}}{4-3} = 2-\sqrt{3}. \]

### Step 4: General term

Continuing this process, we can see that the \(n\)-th term can be expressed as:

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

### Step 5: Write the entire sum

So, the entire sum \(S\) can be rewritten as:

\[ S = (\sqrt{2}-1) + (\sqrt{3}-\sqrt{2}) + (\sqrt{4}-\sqrt{3}) + \ldots + (\sqrt{100}-\sqrt{99}). \]

### Step 6: Simplify the sum

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

\[ S = -1 + \sqrt{100} = -1 + 10 = 9. \]

### Final Answer

Thus, the value of the expression is

\[ \boxed{9}. \]