1975 Canadian MO Problems/Problem 2
A sequence of numbers satisfies
Determine the value of
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.
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