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Difference between revisions of "1989 AHSME Problems/Problem 10"
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== Problem ==
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Consider the sequence defined recursively by <math>u_1=a</math> (any positive number), and <math>u_{n+1}=-1/(u_n+1)</math>, <math>n=1,2,3,...</math> For which of the following values of <math>n</math> must <math>u_n=a</math>?
 
Consider the sequence defined recursively by <math>u_1=a</math> (any positive number), and <math>u_{n+1}=-1/(u_n+1)</math>, <math>n=1,2,3,...</math> For which of the following values of <math>n</math> must <math>u_n=a</math>?
  
 
<math> \mathrm{(A) \ 14 } \qquad \mathrm{(B) \ 15 } \qquad \mathrm{(C) \ 16 } \qquad \mathrm{(D) \ 17 } \qquad \mathrm{(E) \ 18 }  </math>
 
<math> \mathrm{(A) \ 14 } \qquad \mathrm{(B) \ 15 } \qquad \mathrm{(C) \ 16 } \qquad \mathrm{(D) \ 17 } \qquad \mathrm{(E) \ 18 }  </math>
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== Solution ==
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Repeatedly applying the function, and simplifying, we get <cmath>a,\quad-\frac1{a+1},\quad-\frac{a+1}a,</cmath>and then <math>a</math> again. So <math>a</math> must appear at every third term after <math>u_1</math>. The only option given of the form <math>1+3k</math> is <math>\boxed{\mathrm{(C)}\,16}</math>.
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== See also ==
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{{AHSME box|year=1989|num-b=9|num-a=11}} 
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[[Category: Introductory Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 12:48, 3 February 2016

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 (Problems • Answer Key • Resources)
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
All AHSME Problems and Solutions

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