University of South Carolina High School Math Contest/1993 Exam/Problem 24

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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University of South Carolina High School Math Contest/1993 Exam/Problem 24

Problem

Solution