Let `S=Sigma_(n=1)^(999) (1)/((sqrt(n)+sqrt(n+1))(4sqrt(n)+4sqrtn+1))` , then S equals ___________.

Explanation

To solve the problem, we need to evaluate the summation

\[ S = \sum_{n=1}^{999} \frac{1}{(\sqrt{n} + \sqrt{n+1})(4\sqrt{n} + 4\sqrt{n} + 1)}. \]

### Step 1: Simplify the Denominator
We start by simplifying the denominator:

\[ (\sqrt{n} + \sqrt{n+1})(4\sqrt{n} + 4\sqrt{n} + 1) = (\sqrt{n} + \sqrt{n+1})(5\sqrt{n}). \]

### Step 2: Rationalize the Expression
Next, we can rationalize the expression by multiplying the numerator and denominator by \((\sqrt{n+1} - \sqrt{n})\):

\[ S = \sum_{n=1}^{999} \frac{\sqrt{n+1} - \sqrt{n}}{(\sqrt{n+1} - \sqrt{n})(\sqrt{n} + \sqrt{n+1})(5\sqrt{n})}. \]

### Step 3: Apply the Difference of Squares
Using the difference of squares, we can simplify the denominator:

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

Thus, we have:

\[ S = \sum_{n=1}^{999} \frac{\sqrt{n+1} - \sqrt{n}}{5\sqrt{n}}. \]

### Step 4: Rearranging the Summation
Now we can separate the summation:

\[ S = \frac{1}{5} \sum_{n=1}^{999} (\sqrt{n+1} - \sqrt{n}). \]

### Step 5: Telescoping Series
The series is telescoping, which means that most terms will cancel out:

\[ S = \frac{1}{5} \left( \sqrt{1000} - \sqrt{1} \right). \]

### Step 6: Calculate the Values
Calculating the square roots gives us:

\[ S = \frac{1}{5} (10 - 1) = \frac{9}{5}. \]

### Final Answer
Thus, the value of \( S \) is:

\[ \boxed{\frac{9}{5}}. \]