Example 1. Find the sum of the arithmetic sequence $a_{1}, a_{1}+d, \cdots$, $a_{1}+(n-1) d$.
$$
\left.\begin{array}{l}
a_{1}+d+d+\cdots+d \\
a_{1}+d+d+\cdots+d \\
a_{1}+d+d+\cdots+d \\
\cdots \cdots \cdots \cdots \cdots \cdots+\cdots \\
a_{1}+\underbrace{d+d+\cdots+d}_{n-1}
\end{array}\right\}
$$

Except for the 1st column, the number of $d$ in the upper right triangle is the same as in the lower left triangle, therefore
$$
\begin{array}{c}
S_{n}=a_{1}+\left(a_{1}+d\right)+\cdots \\
+\left[a_{1}+(n-1) d\right]=n a_{1}+\frac{(n-1) n}{2} d
\end{array}
$$