Simplify `P=1/(2sqrt(1)+sqrt(2))+1/(3sqrt(2)+2sqrt(3))+....+1/(100sqrt99+99sqrt100`

Options

A. `(1)/(10)`
B. `(3)/(10)`
C. `(9)/(10)`
D. `(1)/(2)`

Answer

To simplify the expression

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

we will follow these steps:

### Step 1: Simplifying the first term
Consider the first term:

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

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

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

Calculating the denominator:

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

Thus, we have:

\[ \frac{2 - \sqrt{2}}{2} = 1 - \frac{\sqrt{2}}{2}. \]

### Step 2: Simplifying the second term
Now, consider the second term:

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

Again, we multiply by the conjugate:

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

Calculating the denominator:

\[ (3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6. \]

Thus, we have:

\[ \frac{3\sqrt{2} - 2\sqrt{3}}{6} = \frac{1}{2}\sqrt{2} - \frac{1}{3}\sqrt{3}. \]

### Step 3: Continuing the pattern
Continuing this process for each term, we observe a pattern:

\[ \frac{1}{n\sqrt{n-1} + (n-1)\sqrt{n}} \rightarrow \text{results in a similar form}. \]

### Step 4: Writing the general term
The general term can be expressed as:

\[ \frac{\sqrt{n-1}}{n(n-1)} - \frac{\sqrt{n}}{n(n)}. \]

### Step 5: Summing the series
Now we can sum the series:

\[ P = \left(1 - \frac{\sqrt{2}}{2}\right) + \left(\frac{1}{2}\sqrt{2} - \frac{1}{3}\sqrt{3}\right) + \ldots + \left(\frac{\sqrt{99}}{99} - \frac{\sqrt{100}}{100}\right). \]

Notice that this is a telescoping series where many terms will cancel out:

\[ P = 1 - \frac{\sqrt{100}}{100} = 1 - 1 = 0. \]

### Step 6: Final simplification
The last term simplifies to:

\[ P = 1 - \frac{10}{100} = 1 - 0.1 = 0.9 = \frac{9}{10}. \]

Thus, the final answer is:

\[ \boxed{\frac{9}{10}}. \]