Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 24"
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== Problem == | == Problem == | ||
| + | If <math>f(x) = \frac{1 + x}{1 - 3x}, f_1(x) = f(f(x)), f_2(x) = f(f_1(x)),</math> and in general <math>f_n(x) = f(f_{n-1}(x)),</math> then <math>f_{1993}(3)=</math> | ||
| − | <center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } </math></center> | + | <center><math> \mathrm{(A) \ }3 \qquad \mathrm{(B) \ }1993 \qquad \mathrm{(C) \ }\frac 12 \qquad \mathrm{(D) \ }\frac 15 \qquad \mathrm{(E) \ } -2^{-1993} </math></center> |
== Solution == | == Solution == | ||
| + | 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>. | ||
== See also == | == See also == | ||
* [[University of South Carolina High School Math Contest/1993 Exam]] | * [[University of South Carolina High School Math Contest/1993 Exam]] | ||
Revision as of 19:40, 22 July 2006
Problem
If 
and in general 
then 
Solution
Notice that for each 
where 
is odd, the value is 
and for each value of where is even, the value is 
. It follows that the answer is .
