The summation of series `sum_(r=1->99) 1/(sqrt(r+1)+sqrtr)` is:

Options

A. 10
B. 9
C. 1
D. 6

Answer

To solve the summation of the series

\[ \sum_{r=1}^{99} \frac{1}{\sqrt{r+1} + \sqrt{r}}, \]

we will rationalize the denominator and simplify the expression step by step.

### Step 1: Rationalize the Denominator

We start with the term

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

To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{r+1} - \sqrt{r}\):

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

### Step 2: Simplify the Denominator

The denominator simplifies as follows:

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

Thus, we have:

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

### Step 3: Rewrite the Summation

Now, we can rewrite the summation:

\[ \sum_{r=1}^{99} \left( \sqrt{r+1} - \sqrt{r} \right). \]

### Step 4: Evaluate the Summation

This is a telescoping series. When we expand the summation, we get:

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

Notice that all intermediate terms cancel out:

\[ -\sqrt{1} + \sqrt{100}. \]

### Step 5: Calculate the Final Result

The remaining terms after cancellation are:

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

Thus, the final result of the summation is:

\[ \sum_{r=1}^{99} \frac{1}{\sqrt{r+1} + \sqrt{r}} = 9. \]

### Final Answer

The answer is \(9\).

---