1989 AHSME Problems/Problem 10

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Problem

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 $u_n=a$?

$\mathrm{(A) \ 14 } \qquad \mathrm{(B) \ 15 } \qquad \mathrm{(C) \ 16 } \qquad \mathrm{(D) \ 17 } \qquad \mathrm{(E) \ 18 }$

Solution

Repeatedly applying the function, and simplifying, we get \[a,\quad-\frac1{a+1},\quad-\frac{a+1}a,\]and then $a$ again. So $a$ must appear at every third term after $u_1$. The only option given of the form $1+3k$ is $\boxed{\mathrm{(C)}\,16}$.

See also

1989 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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