5. (10 points) 30 students are lined up in ascending order of height, with the height difference between adjacent students being the same. The sum of the heights of the first 10 students is 12.5 meters, and the sum of the heights of the first 20 students is 26.5 meters. What is the sum of the heights of these 30 students in meters?
【Solution】Solution: According to the analysis, the heights of 30 students form an arithmetic sequence. Let the height of the $n$-th student be $a_{n}$, and the sum of the heights of the first $n$ students be $S_{n}$. Then, $S_{10}=12.5$ meters, $S_{20}=26.5$ meters. According to the properties of an arithmetic sequence,
$$
S_{10}=a_{1}+a_{2}+\cdots a_{10} ; \quad S_{20}-S_{10}=a_{11}+a_{12}+\cdots+a_{20} ; \quad S_{30}-S_{20}=a_{21}+a_{22}+\cdots+a_{30} .
$$

It is easy to know that $S_{10} ; S_{20}-S_{10} ; S_{30}-S_{20}$ is an arithmetic sequence, so $S_{20}-S_{10}=26.5-12.5=14$ meters;
$$
\begin{array}{l}
S_{30}-S_{20}=S_{10}+2 \times(14-12.5)=12.5+3=15.5 \text { meters; } \\
\Rightarrow S_{30}=S_{20}+15.5=26.5+15.5=42 \text { meters. }
\end{array}
$$
$\therefore$ The sum of the heights of these 30 students is 42 meters.
The answer is: 42 meters.