1975 Canadian MO Problems/Problem 2
Problem 2
A sequence of numbers satisfies
(i)
(ii)
Determine the value of
Solution
I claim with a proof by induction. First we can use partial fraction decomposition to rewrite as . We have We can set coefficients equal, and . Now,
Base Case: If , then So, when .
Inductive Step: Suppose conclusion is true for , such that we have We also have Add to both sides. The left side becomes which is a telescoping series equal to . Now, we have We have thus the conclusion being true for , implies that it holds for , so our induction is complete.
1975 Canadian MO (Problems) | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 3 |