[Solution] Let the shorter base, height, and longer base of the trapezoid be $a-d, a, a+d$ respectively. Then $k=S_{\text {trapezoid }}=\frac{1}{2}[(a-d)+(a+d)] a=a^{2}$. By the arbitrariness of $a$, we know that $(A),(B),(C),(D)$ are all incorrect. Therefore, the answer is $(E)$.
$18 \cdot 120$ is the short base, height, and long base of a trapezoid, which form an arithmetic sequence. Let $k$ be the area of this trapezoid, then
(A) $k$ must be an integer.
(B) $k$ must be a fraction.
(C) $k$ must be an irrational number.
(D) $k$ must be an integer or a fraction.
(E) None of (A), (B), (C), (D) is correct.
(18th American High School Mathematics Examination, 1967)