1. Calculate the value of the expression

$$
S=\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}
$$

where $a_{1}>0, a_{2}>0, \ldots, a_{n}>0$ and $a_{1}, a_{2}, \ldots, a_{n}$ form an arithmetic progression.
Solution. We have:

$$
\begin{aligned}
S & =\frac{\sqrt{a_{1}}-\sqrt{a_{2}}}{a_{1}-a_{2}}+\frac{\sqrt{a_{2}}-\sqrt{a_{3}}}{a_{2}-a_{3}}+\ldots+\frac{\sqrt{a_{n-1}}-\sqrt{a_{n}}}{a_{n-1}-a_{n}} \\
& =-\frac{1}{d}\left(\sqrt{a_{1}}-\sqrt{a_{2}}+\sqrt{a_{2}}-\sqrt{a_{3}}+\ldots+\sqrt{a_{n-1}}-\sqrt{a_{n}}\right) \\
& =\frac{1}{d}\left(\sqrt{a_{n}}-\sqrt{a_{1}}\right)=\frac{n-1}{\sqrt{a_{n}}+\sqrt{a_{1}}}
\end{aligned}
$$