Let `S=(sqrt(1))/(1+sqrt1+sqrt(2))+sqrt(2)/(1+sqrt(2)+sqrt(3))+(sqrt(3))/(1+sqrt(3)+sqrt(4))+...+(sqrt(n))/(1+sqrt(n)+(sqrtn+1))=10`
Then find the value of n.

Explanation

To solve the problem, we need to evaluate the series given by:

\[ S = \sum_{r=1}^{n} \frac{\sqrt{r}}{1 + \sqrt{r} + \sqrt{r+1}} = 10 \]

### Step 1: Simplifying the General Term

The general term of the series can be simplified. We have:

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

To simplify this, we can rationalize the denominator. We multiply the numerator and the denominator by the conjugate of the denominator:

\[ 1 + \sqrt{r} - \sqrt{r+1} \]

Thus, we have:

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

### Step 2: Simplifying the Denominator

The denominator simplifies as follows:

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

So, the general term becomes:

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

### Step 3: Writing the Series

Now, we can express the series \(S\):

\[ S = \sum_{r=1}^{n} \frac{1 + \sqrt{r} - \sqrt{r+1}}{2} \]

This can be split into three separate sums:

\[ S = \frac{1}{2} \sum_{r=1}^{n} 1 + \frac{1}{2} \sum_{r=1}^{n} \sqrt{r} - \frac{1}{2} \sum_{r=1}^{n} \sqrt{r+1} \]

### Step 4: Evaluating Each Sum

The first sum is straightforward:

\[ \sum_{r=1}^{n} 1 = n \]

The second sum can be expressed as:

\[ \sum_{r=1}^{n} \sqrt{r} \quad \text{and} \quad \sum_{r=1}^{n} \sqrt{r+1} = \sum_{r=2}^{n+1} \sqrt{r} \]

Thus, we can rewrite \(S\):

\[ S = \frac{1}{2} \left( n + \sum_{r=1}^{n} \sqrt{r} - \sum_{r=2}^{n+1} \sqrt{r} \right) \]

The terms \(\sum_{r=1}^{n} \sqrt{r}\) and \(\sum_{r=2}^{n+1} \sqrt{r}\) will cancel out, leaving:

\[ S = \frac{1}{2} \left( n + \sqrt{1} - \sqrt{n+1} \right) = \frac{1}{2} \left( n + 1 - \sqrt{n+1} \right) \]

### Step 5: Setting Up the Equation

We know that \(S = 10\):

\[ \frac{1}{2} (n + 1 - \sqrt{n+1}) = 10 \]

Multiplying both sides by 2 gives:

\[ n + 1 - \sqrt{n+1} = 20 \]

Rearranging this gives:

\[ n - \sqrt{n+1} = 19 \]

### Step 6: Substituting and Solving

Let \(x = \sqrt{n+1}\). Then \(n = x^2 - 1\). Substituting this into the equation gives:

\[ x^2 - 1 - x = 19 \]

This simplifies to:

\[ x^2 - x - 20 = 0 \]

### Step 7: Factoring the Quadratic

Factoring the quadratic:

\[ (x - 5)(x + 4) = 0 \]

Thus, \(x = 5\) or \(x = -4\). Since \(x\) must be non-negative, we take \(x = 5\).

### Step 8: Finding \(n\)

Now substituting back to find \(n\):

\[ \sqrt{n+1} = 5 \implies n + 1 = 25 \implies n = 24 \]

### Final Answer

The value of \(n\) is:

\[ \boxed{24} \]