The sum of `1/(sqrt(2)+1) + 1/(sqrt(3) + sqrt(2)) + 1/(sqrt(4) + sqrt(3)) +.....1/(sqrt(100) + sqrt(99))` is equal to:

Options

A. 9
B. 10
C. 11
D. 0

Answer

To solve the problem, we need to evaluate the sum:

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

### Step 1: Rationalizing Each Term

We will rationalize each term in the sum. The general term can be expressed as:

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

To rationalize, we multiply the numerator and the denominator by the conjugate of the denominator:

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

### Step 2: Rewrite the Sum

Now we can rewrite the entire sum \( S \):

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

### Step 3: Notice the Telescoping Nature

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

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

### Step 4: Calculate the Final Value

Now we can compute the final value:

\[ S = -1 + 10 = 9 \]

Thus, the sum is:

\[ \boxed{9} \]