1973 Canadian MO Problems/Problem 7
Revision as of 19:02, 4 December 2015 by Pi3point14 (talk | contribs)
Problem
Observe that $\frac{1}{2}+<cmath>\frac{1}{2};</cmath>\quad \frac{1}{2}=\frac{1}{3}+\frac{1}{6};\quad \frac{1}{3}=\frac{1}{4}+\frac{1}{12};\qu...$ (Error compiling LaTeX. Unknown error_msg) State a general law suggested by these examples, and prove it.
Prove that for any integer greater than there exist positive integers and such that
Solution
We see that:
We prove this by induction. Let Base case: Therefore, is true. Now, assume that is true for some . Then:
Thus, by induction, the formula holds for all
Incomplete
See also
1973 Canadian MO (Problems) | ||
Preceded by Problem 6 |
1 • 2 • 3 • 4 • 5 | Followed by Problem 1 |