10.10 The sequence $\left\{u_{n}\right\}, u_{1}=a$ (where $a$ is any positive number), $u_{n+1}=-\frac{1}{u_{n}+1}$, $n=1,2,3, \cdots$, which of the following values of $n$ can make $u_{n}=a$?
(A) 14.
(B) 15.
(C) 16.
(D) 17.
(E) 18.
(40th American High School Mathematics Examination, 1989)
[Solution] Given $u_{1}=a, u_{2}=-\frac{1}{a+1}$,
$$
u_{3}=-\frac{1}{1-\frac{1}{a+1}}=-1-\frac{1}{a}, \quad u_{4}=-\frac{1}{1-1-\frac{1}{a}}=a \text {. }
$$
$\therefore\left\{u_{n}\right\}$ has a period of 3, and since $3 \mid(16-1)$, the answer is $(C)$.