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Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 24"
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Notice that for each <math>f_{n}(3)</math> where <math>n</math> is odd, the value is <math>1/5</math> and for each value of <math>f_{n}(3)</math> where <math>n</math> is even, the value is <math>3</math>. It follows that the answer is <math>1/5</math>.
 
Notice that for each <math>f_{n}(3)</math> where <math>n</math> is odd, the value is <math>1/5</math> and for each value of <math>f_{n}(3)</math> where <math>n</math> is even, the value is <math>3</math>. It follows that the answer is <math>1/5</math>.
  
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== See also ==
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* [[University of South Carolina High School Math Contest/1993 Exam]]
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* [[University of South Carolina High School Math Contest/1993 Exam/Problem 23|Previous Problem]]
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* [[University of South Carolina High School Math Contest/1993 Exam/Problem 25|Next Problem]]
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* [[University of South Carolina High School Math Contest/1993 Exam|Back to Exam]]
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[[Category:Intermediate Algebra Problems]]

Revision as of 10:48, 23 July 2006

Problem

If $f(x) = \frac{1 + x}{1 - 3x}, f_1(x) = f(f(x)), f_2(x) = f(f_1(x)),$ and in general $f_n(x) = f(f_{n-1}(x)),$ then $f_{1993}(3)=$

$\mathrm{(A) \ }3 \qquad \mathrm{(B) \ }1993 \qquad \mathrm{(C) \ }\frac 12 \qquad \mathrm{(D) \ }\frac 15 \qquad \mathrm{(E) \ } -2^{-1993}$

Solution

Notice that for each $f_{n}(3)$ where $n$ is odd, the value is $1/5$ and for each value of $f_{n}(3)$ where $n$ is even, the value is $3$. It follows that the answer is $1/5$.

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