$19 \cdot 23$ The interior angles of a convex polygon form an arithmetic sequence. If the smallest angle is $100^{\circ}$ and the largest angle is $140^{\circ}$, then the number of sides of the polygon is
(A) 6.
(B) 8.
(C) 10.
(D) 11.
(E) 12.
(27th American High School Mathematics Examination, 1976)
[Solution] If $n$ is the number of sides of the polygon, then the sum of the interior angles of a convex $n$-sided polygon is $(n-2) 180^{\circ}$. By the formula for the sum of the first $n$ terms of an arithmetic sequence, we have
$$
(n-2) 180=\frac{n}{2}(100+140) \text {, }
$$

Solving for $n$ gives $n=6$.
Therefore, the answer is $(A)$.