To solve the problem, we start by analyzing the general term of the sum and using rationalization to simplify it.

Step 1: Rationalize the general term

The general term is:
[ \frac{1}{\sqrt{k+1} + \sqrt{k}} ]

Multiply numerator and denominator by (\sqrt{k+1} - \sqrt{k}):
[ \frac{1}{\sqrt{k+1}+\sqrt{k}} \times \frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k+1}-\sqrt{k}} = \frac{\sqrt{k+1}-\sqrt{k}}{(k+1)-k} = \sqrt{k+1}-\sqrt{k} ]

Step 2: Telescoping sum

The sum becomes:
[ \sum_{k=1}^{n-1} (\sqrt{k+1}-\sqrt{k}) ]

All intermediate terms cancel out:
[ (\sqrt{2}-\sqrt{1}) + (\sqrt{3}-\sqrt{2}) + ... + (\sqrt{n}-\sqrt{n-1}) = \sqrt{n} - 1 ]

Step 3: Solve for (n)

Given the sum equals 10:
[ \sqrt{n} - 1 = 10 ]
[ \sqrt{n} = 11 ]
[ n = 11^2 = 121 ]

Answer: (\boxed{121})

作者声明:本文包含人工智能生成内容。