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 |