Consider the sequence defined recursively by u_1=a (any positive number), and u_n+1=-(1)/(u_n+1), n=1, 2, 3, …. For which of the following values of n must u_n=a?

1989 AMC 12 Problem 10

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10.

Consider the sequence defined recursively by

u_1=a

(any positive number), and

u_{n+1}=-\frac{1}{u_n+1},

n=1,

2,

3,

\ldots.

For which of the following values of

n

must

u_n=a?

14

15

16

17

18

Compute

u_2,u_3,

and

u_4

symbolically

Look for the period of the transformation

x\mapsto-\frac{1}{x+1}

Direct substitution gives

\begin{aligned} u_2&=-\frac1{a+1},\\ u_3&=-\frac{a+1}{a},\\ u_4&=a. \end{aligned}

The sequence therefore repeats every

3

terms, so

u_n=a

whenever

n\equiv1\pmod3.

Among the choices, only

16

has that form.

Thus the correct answer is C.