2014 AMC 10A 第 8 题

Write each choice as $\frac{n!(n+1)!}{2}$

Only $\frac{n+1}{2}$ needs to be a square

Note that all of these answer choices are of the form $$\dfrac{n!(n + 1)!}{2} = \dfrac{(n!)^2(n + 1)}{2}.$$ We have that $(n!)^2$ is square, so we need $\dfrac{n + 1}{2}$ to be square as well.

This means that $n + 1$ must be twice a perfect square. The only choice we have is $n + 1 = 18,$ which gives us $n = 17.$

Thus, D is the correct answer.