21. Find the value of the positive integer $n$ if
$$
\frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\cdots+\frac{1}{\sqrt{n}+\sqrt{n+1}}=10
$$
21. Ans: 143
Observe that for any positive number $x$,
Thus we have
$$
\begin{aligned}
& \frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\cdots+\frac{1}{\sqrt{n}+\sqrt{n+1}} \\
= & (\sqrt{5}-\sqrt{4})+(\sqrt{6}-\sqrt{5})+\cdots+(\sqrt{n+1}-\sqrt{n}) \\
= & \sqrt{n+1}-\sqrt{4} .
\end{aligned}
$$

Hence, $\sqrt{n+1}-\sqrt{4}=10 \Longrightarrow \sqrt{n+1}=12 \Longrightarrow n+1=144 \Longrightarrow n=143$.