$10 \cdot 21$ The sum of the first 10 terms and the sum of the first 100 terms of an arithmetic sequence are 100 and 10, respectively. What is the sum of the first 110 terms?
(A) 90.
(B) -90.
(C) 110.
(D) -110.
(E) -100.
(31st American High School Mathematics Examination, 1980)
[Solution] The formula for the sum of the first $n$ terms of an arithmetic sequence $S_{n}$ is
$2 S_{n}=n[2 a+(n-1) d]$, where the first term is $a$, and the common difference is $d$.

From the problem, we have
$$
\begin{array}{c}
200=10(2 a+9 d) \text { and } \quad 20=100(2 a+99 d), \\
2 S_{110}=110(2 a+109 d) .
\end{array}
$$

From the first two equations, we get $\quad 2 a+109 d=-2$.
Therefore, $2 S_{110}=110 \cdot(-2)$, so $S_{110}=-110$. Hence, the answer is $(D)$.