1. Given a positive integer $n$, which of the following numbers differs the least from $\left(n+\frac{3}{2}\right)^{2}$ ?
(A) $(n+2)^{2}$
(B) $(n+1)^{2}$
(C) $(n+1)(n+2)$
(D) $n^{2}+2$
(E) $n^{2}+\frac{9}{4}$
Problem 1. The correct answer is (C).

We have $\left(n+\frac{3}{2}\right)^{2}=n^{2}+3 n+\frac{9}{4}$. The value closest to it among the options given is $(n+1)(n+2)=n^{2}+3 n+2$, from which $\left(n+\frac{3}{2}\right)^{2}$ differs by $\frac{1}{4}$ regardless of $n$.