University of South Carolina High School Math Contest/1993 Exam/Problem 5
Problem
Suppose that is a function with the property that for all
and
,
and
What is the value of
?
![$\mathrm{(A) \ }4 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ }6 \qquad \mathrm{(D) \ }7 \qquad \mathrm{(E) \ }8$](http://latex.artofproblemsolving.com/e/7/1/e719d0dae853a231d3f6f4ffaeeb56648c1f073b.png)
Solution
Notice that . Also,
. Thus,
.
In general, , so we have a simple recursive definition for the function
. From here we can see that
for all positive integers
.